ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY

ZEÉV RUDNICK AND PETER SARNAK

1. Introduction

Our goal in this paper is to study the distribution of zeros of the Riemann zeta function as well as of more general L-functions^{1}. According to conjectures of Langlands [14], the most general L-function is that attached to an automorphic representation of GL_{N} over a number field, and these in turn should be expressible as products of the “standard” L-functions L(s, π) attached to cuspidal automorphic representations of GL_{m} over the rationals. Such L-functions are therefore believed to be the building blocks for general L-functions and we call them (principal) primitive L-functions of degree m (they do not factor as products of such L-functions)^{2}. For m = 1 these are the Riemann zeta function ζ(s) and Dirichlet L-functions L(s, χ) with χ primitive. For m = 2 the analytic properties and functional equation of such L-functions were investigated by Hecke and Maass, and for m ≥ 3 by Godement and Jacquet [5]. We are interested in the fine structure of the distribution of the non-trivial zeros of such primitive L(s, π). Let ρ^{(π)} = \frac{1}{2} + iγ^{(π)} denote these zeros. To motivate the formulation of our results we begin by assuming the Riemann Hypothesis (RH) for L(s, π), that is that γ^{(π)} ∈ R. We order the γ^{(π)}’s (with multiplicities)

⋯ ≤ γ_{−2}^{(π)} ≤ γ_{−1}^{(π)} < 0 ≤ γ_{1}^{(π)} ≤ γ_{2}^{(π)} …

Partially supported by NSF grants DMS-9400163 and DMS-9102082.

1The reader interested only in the Riemann zeta function ζ(s) should read the paper with L(s, π) replaced by ζ(s) and m = 1 everywhere, in which case the results were announced in [26].

2It is quite plausible that these coincide with the primitive Dirichlet series introduced by Selberg [29] or the “arithmetic Dirichlet series” in Piatetski-Shapiro [24].

2 ZEÉV RUDNICK AND PETER SARNAK

The number of γ’s in an interval [T, T + 1] is asymptotic to \frac{m}{2π} log T as T → ∞ (see (2.11)). It follows that the numbers γ̃_{j}^{(π)} = \frac{m}{2π}γ_{j} log |γ_{j}| have unit mean spacing. The problem is to understand the statistical nature of the sequence γ̃_{j}^{(π)}: Do they come down randomly (Poisson process) or do they follow a more revealing distribution?

In the case of the Riemann zeta function, following the original calculation by Montgomery [20] of the pair correlation (see below) and the extensive numerical calculations of Odlyzko [21], [22], it is now well accepted (but far from proven) that the consecutive spacings follow the Gaussian Unitary Ensemble (GUE) distribution from Random Matrix Theory. That is if δ_{n} = γ̃_{n+1} − γ̃_{n} are the normalized spacings, then for any nice function on (0, ∞) one expects

\frac{1}{N} ∑_{n≤N} f(δ_{n}) → ∫_{0}^{∞} f(s)P (s)ds  (1.1)

where P (s) is the distribution of consecutive spacings of the eigenvalues of a large random Hermitian matrix. This distribution was determined by Gaudin and Mehta [18] and is given as follows: P (s) = \frac{d^{2}E}{ds^{2}} (s) where E(s) = det(I − Q_{s}), Q_{s} being the trace class operator on L^{2}(−1, 1) whose kernel is Q_{s}(ξ, η) = \frac{sin πs(ξ−η)/2}{π(ξ−η)}. The Fredholm determinant defining E(s) converges and it is easy to check that P (s) vanishes to second order at s = 0. This means that unlike a Poisson process, the numbers γ̃_{n} tend to “repel” each other (this is often referred to as “level repulsion”). The graph of P (s) and its comparison with Odlyzko’s computation for zeros near the 10^{20}-th are depicted in Figure 1.

For Dirichlet L-functions, the picture is similar and Rumely [27] has carried out analogous numerical experiments. Recently Hejhal [6] has succeeded in computing the 3-level correlation function for zeros of ζ(s), assuming RH and in a restricted range, similar to the assumption and restriction used by Montgomery.

The consecutive level spacing distribution is determined by the nlevel correlation functions for all n ≥ 2 [28]. The main result of this paper is the computation of the general n-level correlation function for the zeros of a primitive principal L-function (also in a restricted

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 3

range). We show that the answer is universal and is precisely the one predicted by Dyson’s computations for the GUE model [3]. To define the n-level correlations suppose that as above we have a set B_{N} of N numbers γ̃_{1} ≤ ⋯ ≤ γ̃_{N}. The n-level correlation function measures the correlation between differences of n elements of B_{N}. That is, for a box Q ⊂ R^{n−1} set

R_{n}(B_{N}, Q) = \frac{1}{N} #\{j_{1}, …, j_{n} ≤ N distinct : (γ̃_{j_{1}} − γ̃_{j_{2}}, …, γ̃_{j_{n−1}} − γ̃_{j_{n}}) ∈ Q\}

(1.2)

A technically more convenient way to measure this distribution is to use smooth test functions f(x_{1}, …, x_{n}) satisfying:

TF 1. f(x_{1}, …, x_{n}) is symmetric;

TF 2. f(x + t(1, …, 1)) = f(x) for t ∈ R;

TF 3. f(x) → 0 rapidly as |x| → ∞ in the hyperplane ∑_{j} x_{j} = 0.

The n-level correlation sum R_{n}(B_{N}, f) is defined by

R_{n}(B_{N}, f) = \frac{n!}{N} ∑_{\begin{matrix}S⊂B_{N} \\ |S|=n \end{matrix}} f(S)  (1.3)

Here f(S) = f(a_{1}, …, a_{n}) if S = {a_{1}, …, a_{n}} - since f is symmetric this is well defined. Condition TF 2 asserts that f is a function of the successive differences so that we recover what (1.2) seeks to measure. Condition TF 2 together with the localization TF 3 means that we can think of R_{n}(B_{N}, f) as counting clusters of size n in B_{N}. It turns out that knowing the asymptotic behaviour of R_{n}(B_{N}, f) as N → ∞ is equivalent to knowing that of the smoothed correlations

R_{n}(T, f, h) = ∑_{j_{1},...,j_{n}}^{′} h(\frac{γ_{j_{1}}}{T}) … h(\frac{γ_{j_{n}}}{T})f(\frac{L}{2π} γ_{j_{1}}, …, \frac{L}{2π} γ_{j_{n}}) (1.4)

for a sufficiently rich family of localized cutoff functions h (e.g. of rapid decrease). Here L = m log T, and ∑′ means sum over distinct indices. Note that since h localizes γ to be of order T, the normalization \frac{L}{2π}γ is the same as γ̃.

4 ZEÉV RUDNICK AND PETER SARNAK

As mentioned above, Dyson [3] determined the limiting n-level correlation density W_{n}(x_{1}, …, x_{n}) for the GUE model. He showed it is given by

W_{n}(x_{1}, …, x_{n}) = det(K(x_{i} − x_{j})) ,  K(x) = \frac{sin πx}{πx} (1.5)

W_{n}(x) is a density (though not a probability density) satisfying 0 ≤ W_{n}(x) ≤ 1 with W_{n}(x) = 0 if and only if x_{i} = x_{j} for some i ≠ j, and W_{n}(x) = 1 if and only if x_{i} − x_{j} ∈ Z and x_{i} ≠ x_{j} for all i ≠ j.

Before stating our results we need a technical hypothesis concerning the coefficients of L(s, π). For Re(s) large, write

\frac{L′}{L} (s, π) = − ∑_{n=1}^{∞} \frac{Λ(n)a_{π}(n)}{n^{s}}  (1.6)

where Λ(n) = log p if n = p^{e} is a prime power, and is 0 otherwise. The hypothesis asserts that for any k ≥ 2

∑_{p} \frac{|a_{π}(p^{k}) log p|^{2}}{p^{k}} < ∞  (1.7)

This is a very mild hypothesis. Firstly the general “Ramanujan conjectures” for cusp forms on GL_{m} assert that |a_{π}(p^{k})| ≤ m which yields (1.7) with a lot to spare. Secondly we show in Section 2 that (1.7) is valid for m ≤ 3.

Returning to the n-level correlations we note that if h and f are defined for complex argument and are localized then the sums (1.4) make sense even if we do not assume RH, and we still refer to these as the n-level correlations. As explained above, RH and the GUE model (if it applies) can be used to predict their asymptotic behaviour as T → ∞. Our first result proves that this prediction is correct at least for a restricted class of f’s.

Theorem 1.1. Let π be a cuspidal automorphic representation of GL_{m}/Q. Assume m ≤ 3 or the hypothesis (1.7). Let f satisfy TF 1,2,3 and in addition assume that f̂(ξ) is supported in ∑_{j} |ξ_{j}| < 2/m. Let g ∈ C_{c}^{∞}(R) and h(r) = ∫_{−∞}^{∞} g(u)e^{iru}du (so that h and f are entire).

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 5

Then as T → ∞,

R_{n}(T, f, h) ∼ \frac{m}{2π} T log T ∫_{−∞}^{∞} h(r)^{n}dr ∫_{R^{n}} f(x)W_{n}(x)δ(\frac{x_{1} + ⋯ + x_{n}}{n})dx_{1} … dx_{n}

where δ(x) is the Dirac mass at 0.

If we assume RH for L(s, π) we can relax the smoothness condition on h and in fact choose it to be the characteristic function of an interval. In this way we can prove that the n-level correlations of the zeros are GUE at least for f’s with restricted Fourier transforms. Precisely we deduce from Theorem 1.1

Theorem 1.2. With the assumptions of Theorem 1.1 and also RH for L(s, π),

R_{n}(B_{N}, f) → ∫_{R^{n}} f(x)W_{n}(x)δ(\frac{x_{1} + ⋯ + x_{n}}{n})dx_{1} … dx_{n}

as N → ∞.

Remarks:

1. The restriction ∑_{j} |ξ_{j}| < 2/m is a natural one when m = 1 since in this case it is exactly the region in which the asymptotic behaviour of R_{n}(B_{N}, f) is dominated by the contributions from all the multidiagonals (see Section 3). Beyond this region a saturation takes effect and the diagonals no longer dominate. For ζ(s) this region is also distinguished by being the range in which the pole at s = 1 contributes only terms of lower order to R_{n}(B_{N}, f). In the case of ζ(s) and n = 2, Theorem 1.2 coincides with the result of Montgomery [20]. For m > 1 the restriction ∑_{j=1}^{n} |ξ_{j}| < 2/m is no longer natural in the sense that we expect the diagonals to continue being the dominant terms as long as ∑_{j=1}^{n} |ξ_{j}| < 2. The difference and heuristic reasoning leading to this is given at the end of Section 3. In all cases we conjecture the complete universality of the n-level correlations - that is to say that Theorems 1.1 and 1.2 hold without any restrictions on the support of f̂. It would be very interesting to check numerically the level spacing distribution for the various types of primitive L-functions of

6 ZEÉV RUDNICK AND PETER SARNAK

degree m = 2 (e.g. of CM type, general type, holomorphic and non-holomorphic).

2. The condition that L(s, π) be primitive (i.e. coming from a cuspidal π over Q) is crucial. Firstly if for example we look at L(s) = ζ(s)^{2} then clearly the distribution of the zeros of L(s) will be GUE with multiplicity two. However, also in the case L(s) = L(s, π_{1})L(s, π_{2}), with π_{1} ≠ π_{2} (e.g. the Dedekind zeta function of a quadratic extension of Q), the distribution will not be GUE. The reason is that one can by these methods easily see that the zeros of distinct primitive L-functions are uncorrelated - so to speak are unaware of each others’ existence. As a consequence the zeros of L(s) will not exhibit the “level repulsion” characteristic of the GUE distribution. Indeed the natural conjecture here is that the zeros of L(s) = L(s, π_{1})L(s, π_{2}) will follow the distribution of the superposition of two GUE’s [18]. This clarifies the role played by the primitive L-functions in understanding the distribution of zeros of the general L-function.

3. The universality (in π) of the distribution of zeros of L(s, π) is somewhat surprising. The reason being that the distribution of the coefficients a_{π}(p) in (1.6), as p runs over primes, is not universal. For example, for degree two primitive L-functions there are two conjectured possible limiting distributions for the a_{π}(p)’s: Sato-Tate or uniform distribution (with a Dirac mass term) [30]. As the degree increases, the number of possible limit distributions increases rapidly. However it is a consequence of the theory of the Rankin-Selberg L-functions (developed by Jacquet, Piatetski-Shapiro and Shalika [8] for m ≥ 3) that all these limiting distributions have the same second moment (at least under hypothesis (1.7)). It is the universality of the second moment that is eventually responsible for the universality in Theorems 1.1 and 1.2. For the case of pair correlation (n = 2) this is reasonably evident; for n > 2 it was (at least for us) unexpected and it has its roots in a

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 7

key feature of “diagonal pairings” that emerges as the main term in the asymptotics of R_{n}(T, f, h) (see Section 3).

To end the Introduction, we outline our proof of Theorems 1.1 and 1.2. As in [20] (and indeed in all work on zeros of L-functions), we use some version of Riemann’s “Explicit Formula” relating sums over zeros to sums over primes. One technical novelty lies in our means of using the Explicit Formula which results in an integrated and smoothed version in Theorem 1.1. This allows us to avoid appealing to RH and also considerably facilitates the computation of the n-level correlation functions. Theorem 1.2 is easily recovered From the smooth version. The advantage is that the multi-dimensional sums over primes that arise can be analyzed by rather “soft” means (e.g. no large sieve inequalities are needed). What emerges as the main term are contributions from diagonal pairs. The combinatorics relating this to what is predicted by GUE, viz. the determinant (1.5) are non-trivial; after all, at some point we have to see this determinant emerge from the theory of primes. The marriage comes from the pairing structure mentioned earlier (which in turn stems from unique factorization) and the cycle structure of the determinant (1.5) - this being encoded in the identity of Theorem 4.1 and Proposition 4.3. This combinatorial analysis is described in Section 4. A crucial ingredient is the combinatorial method of F. Spitzer [34]. In Section 2 we collect various facts about principal L-functions, local factors and Rankin-Selberg L-functions as well as Ramanujantype bounds that will be needed. The use of the Explicit Formula to convert the problem to sums over primes is carried out in Section 3. The Appendix contains the calculations of the Rankin-Selberg local factors at the ramified places.

8 ZEÉV RUDNICK AND PETER SARNAK

2. Background on L-functions

2.1. Principal L-functions. This section is devoted to reviewing some more or less standard facts about automorphic L-functions on GL_{m}. Our emphasis in places will be on the higher rank theory (m ≥ 3), the results for Dirichlet L-functions being well known and the case m = 2 is by now also classical. For definitions and proofs of various statements below see Jacquet’s article [7]. We have also included an Appendix in which proofs are given of some facts that we could not find either explicitly stated or proved in the literature.

Let π = ⊗_{p}π_{p} be an irreducible cuspidal automorphic representation of GL_{m}/Q. For normalization purposes, we assume that π is unitary, by which we mean that the central character ω_{π} of π is unitary. To π one associates an Euler product L(s, π) = ∏_{p} L(s, π_{p}) given by a product of local factors. Outside of a finite set of primes S_{π}, π_{p} is unramified and we can associate to π_{p} a semi-simple conjugacy class {A_{π}(p)} ∈ GL_{m}(C). Such a conjugacy class is parametrized by its eigenvalues α_{π}(j, p), j = 1, …, m. The local factors L(s, π_{p}) for the unramified primes are given by

L(s, π_{p}) = det(I − p^{−s}A_{π}(p))^{−1} = ∏_{j=1}^{m}(1 − α_{π}(j, p)p^{−s})^{−1} (2.1)

At the ramified finite primes, the local factors are best described by the Langlands parameters of π_{p} (see the Appendix). They are of the form L(s, π_{p}) = P_{p}(p^{−s})^{−1}, where P_{p}(x) is a polynomial of degree at most m, and P_{p}(0) = 1. We will find it convenient in this case too to write the local factors in the form (2.1), with the convention that we now allow some of the α’s to be zero.

The local constituents π_{p} of a cuspidal π as above are generic [23], [33]. Using local methods, Jacquet and Shalika [10] show that a generic π_{p} satisfies

|α_{π}(j, p)| < p^{1/2}  (2.2)

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 9

The general “Ramanujan conjectures” for cuspidal automorphic π on GL_{m} assert that for p unramified, |α_{π}(j, p)| = 1. This is known for certain π (e.g. on GL_{2} corresponding to holomorphic forms, due to Deligne) but certainly not in general. We will derive a slightly sharper estimate than (2.2) for all p < ∞. In the Appendix it is shown by a well known global argument that for any p < ∞,

|α_{π}(j, p)| ≤ p^{\frac{1}{2} − \frac{1}{m^{2}+1}}  (2.3)

There is also an archimedean local factor L(s, π_{∞}). Again it is best described in terms of the Langlands parameters of π_{∞} - see the Appendix. For now it suffices to note that L(s, π_{∞}) can be written as a product of m Gamma factors:

L(s, π_{∞}) = ∏_{j=1}^{m} Γ_{R}(s + μ_{π}(j))  (2.4)

where Γ_{R}(s) = π^{−s/2}Γ(\frac{s}{2}) and \{μ_{π}(j)\} is a set of m numbers associated to π_{∞}. They satisfy the analogue of (2.2)

Re(μ_{π}(j)) > −\frac{1}{2}  (2.5)

We refer to the Appendix for a discussion of (2.5) and of the analogue of (2.3).

2.2. The functional equation. With all the local factors defined, we can turn to the functional equation. Firstly from (2.2) it is clear that

L(s, π) = ∏_{p<∞} L(s, π_{p})  (2.6)

converges absolutely at least for Re s > 3/2. Set

Φ(s, π) = L(s, π_{∞})L(s, π)  (2.7)

Associated to π is its contragredient π̃, which is itself an irreducible cuspidal automorphic representation. For any p ≤ ∞, π̃_{p} is equivalent to the complex conjugate \overline{π_{p}} [4] and hence

\begin{aligned}\{απ̃(j, p)\} & = \{\overline{α_{π}(k, p)}\} \\ \{μπ̃(j)\} & = \{\overline{μ_{π}(j)}\} \end{aligned}  (2.8)

10 ZEÉV RUDNICK AND PETER SARNAK

The basic analytic result, proven by Godement-Jacquet [5], [7] is that Φ(s, π) extends to an entire function (except in the case of ζ(s) which has a simple pole at s = 1). Moreover Φ(s, π) is bounded in vertical strips and satisfies a functional equation

\begin{aligned}Φ(s, π) & = ϵ(s, π)Φ(1 − s, π̃) \\ ϵ(s, π) & = τ(π)Q_{π}^{−s} \end{aligned}  (2.9)

where Q_{π} > 0 is the conductor of π. It is a positive integer with prime factors in S_{π} [9], and τ(π) ∈ C^{∗}. We note that Qπ̃ = Q_{π} and τ(π)τ(π̃) = Q_{π}.

The zeros of Φ(s, π) will be denoted by ρ_{π} and by definition are the “non-trivial” zeros of L(s, π). The non-trivial zeros of L(s, π̃) are related to those of L(s, π) via s ↦ 1 − s. The analogue of the Riemann Hypothesis for L(s, π) is that Re(ρ_{π}) = 1/2. In as much as Φ(s, π) is of order one and the real parts of of the zeros are constrained to lie in a strip, it follows that the counting function

N_{π}(T) := #{ρ_{π} : | Im ρ_{π}| < T }  (2.10)

satisfies N(T) = O(T^{1+ϵ}) for all ϵ > 0. A standard winding number argument [2] shows that the Gamma factors in L(s, π_{∞}) control the number of zeros, in fact

N_{π}(T) ∼ \frac{m}{π}T log T  (2.11)

2.3. An explicit formula. For Re s > 3/2 we may take the logarithmic derivative of (2.6). This yields

\frac{L′}{L} (s, π) = − ∑_{n=1}^{∞} \frac{Λ(n)a_{π}(n)}{n^{s}}  (2.12)

where Λ(n) = log p, if n = p^{k} and 0 otherwise, while

a_{π}(p^{k}) = ∑_{j=1}^{m} α_{π}(p, j)^{k}  (2.13)

Note that

aπ̃(n) = \overline{a_{π}(n)}  (2.14)

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 11

It will be convenient to set

c_{π}(n) = Λ(n)a_{π}(n)  (2.15)

We recast the information in the Euler product and functional equation in terms of an explicit relation between the zeros ρ_{π} and the a_{π}(p^{k}). Such relations go by the name of “explicit formulae”; the one we use is a smooth version of Riemann’s original formula [25].

Proposition 2.1 (The Explicit Formula). Let g ∈ C_{c}^{∞}(R) be a smooth compactly supported function, and let h(r) = ∫_{−∞}^{∞} g(u)e^{iru}du. Write ρ_{π} = 1/2 + iγ_{π}. Then

∑ h(γ_{π}) = δ(π)\{h(−\frac{i}{2}) + h(\frac{i}{2})\}

+ \frac{1}{2π} ∫_{−∞}^{∞} h(r) (log Q_{π} + ∑_{j=1}^{m} \frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + μ_{π}(j) + ir) + \frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + \overline{μ_{π}(j)} − ir)) dr

− ∑_{n=1}^{∞} (\frac{c_{π}(n)}{\sqrt{n}}g(log n) + \frac{\overline{c_{π}(n)}}{\sqrt{n}}g(− log n))

(2.16)

where δ(π) = 1 if π corresponds to ζ(s), and 0 otherwise.

Proof. Set H(s) := h((s − 1/2)/i), and consider the integral

I = \frac{∞}{∈π⟩} ∫_{Re ∫=∈} \frac{Φ′}{Φ} (∫, π)H(∫)⌈∫  (2.17)

Now H(s) is rapidly decreasing in Im s and is entire, so that the integral converges absolutely, and all contour shifts below are legitimate. Φ′/Φ has simple poles at the zeros of Φ(s, π) with residues the multiplicity of the zero (and in the case of ζ(s) a simple pole with residue −1 at the poles s = 0, 1). Shifting the contour in (2.17) to Re s = −1 we have

I = −δ(π)\{⟨(−\frac{⟩}{∈}) + ⟨(\frac{⟩}{∈})\} + ∑ ⟨(γ_{π}) + \frac{∞}{∈π⟩} ∫_{Re ∫=−∞} \frac{Φ′}{Φ} (∫, π)H(∫)⌈∫

where the sum is over the zeros each counted with its multiplicity. The functional equation (2.9) gives

\frac{Φ′}{Φ} (s, π) = − log Q_{π} − \frac{Φ′}{Φ} (1 − s, π̃)

12 ZEÉV RUDNICK AND PETER SARNAK

Using this and changing variables gives

I = −δ(π)\{⟨(−\frac{⟩}{∈}) + ⟨(\frac{⟩}{∈})\} + ∑ h(γ_{π}) − \frac{1}{2πi} ∫_{Re s=2} log Q_{π}H(s)ds

− \frac{1}{2πi} ∫_{Re s=2} \frac{Φ′}{Φ} (s, π̃)H(1 − s)ds

or:

∑ h(γ_{π}) = δ(π)\{h(−\frac{i}{2}) + h(\frac{i}{2})\} − \frac{1}{2πi} ∫_{Re s=2} log Q_{π}H(s)ds

+ \frac{1}{2πi} ∫_{Re s=2} \frac{Φ′}{Φ} (s, π)H(s)ds + \frac{1}{2πi} ∫_{Re s=2} \frac{Φ′}{Φ} (s, π̃)H(1 − s)ds

Using Φ(s, π) = L(s, π_{∞})L(s, π) we get

\frac{1}{2πi} ∫_{Re s=2} \frac{Φ′}{Φ} (s, π)H(s)ds = \frac{1}{2πi} ∫_{Re s=2} (∑_{j=1}^{m} \frac{Γ′_{R}}{Γ_{R}} (s + μ_{π}(j)) + \frac{L′}{L} (s, π)) H(s)ds

Now shifting the contour of integration to Re s = 1/2

\frac{1}{2πi} ∫_{Re s=2} ∑_{j=1}^{m} \frac{Γ′_{R}}{Γ_{R}} (s + μ_{π}(j))H(s)ds = \frac{1}{2π} ∫_{−∞}^{∞} ∑_{j=1}^{m} \frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + ir + μ_{π}(j))h(r)dr

That no poles are picked up on shifting the contour from Re s = 2 to Re s = 1/2 is equivalent to the inequality (2.5).

Thirdly, using (2.12) we have

\frac{1}{2πi} ∫_{Re s=2} \frac{L′}{L} (s, π)H(s)ds = − ∑_{n=1}^{∞} \frac{c_{π}(n)}{\sqrt{n}} \frac{1}{2π} ∫_{−∞}^{∞} h(r)e^{−ir log n}dr

= − ∑_{n=1}^{∞} \frac{c_{π}(n)}{\sqrt{n}}g(log n)

We do the same for the integral involving Φ′/Φ(s, π̃) and use (2.14) and (2.8). Collecting the terms gives the explicit formula of the Proposition.

2.4. Rankin-Selberg convolutions. A crucial ingredient in Section 3 is the asymptotic behaviour of the mean square of c_{π}(n) = Λ(n)a_{π}(n). To determine the asymptotics we will need the Rankin-Selberg Lfunction. Its general theory has been developed by Jacquet, Piatetski-Shapiro and Shalika [8], and more recently by Shahidi [32] and Mœglin-Waldspurger [19]. For cuspidal automorphic representations π on GL_{m}, π′ on GL_{m′}, the Rankin-Selberg L-function L(s, π × π′) is defined as

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 13

a product of local factors L(s, π × π′) = ∏_{p} L(s, π_{p} × π′_{p}). Initially it is seen to be absolutely convergent for Re s ≫ 1 but in the end one finds this to be so in Re s > 1. For primes p where both π_{p} and π′_{p} are unramified, the local factor is given in terms of the corresponding semi-simple conjugacy classes A_{π}(p), A′_{π}(p) (2.1) by

L(s, π_{p} × π′_{p}) = det(I − p^{−s}A_{π}(p) ⊗ A_{π′}(p))^{−1} = ∏_{j,k}(1 − α_{π}(p, j)α_{π′}(p, k)p^{−s})^{−1} (2.18)

The local factors for ramified primes will be described in the Appendix. They are of the form P_{p}(p^{−s})^{−1}, where P_{p}(x) is a polynomial of degree at most mm′ with P (0) = 1. At infinity the local factor is of the form ∏_{j=1}^{mm′} Γ_{R}(s + μ_{π×π′}(j)). If π_{∞} and π′_{∞} are unramified then

{μ_{π×π′}(j)} = {μ_{π}(j) + μ_{π′}(k)}

See the Appendix for a description of the general case.

For us the case of most interest is π′ = π̃. In this case we see from (2.8) and (2.18) that for π_{p} unramified,

\begin{aligned}log L(s, π_{p} × π̃_{p}) & = ∑_{j,k} ∑_{ν=1}^{∞} \frac{(α_{π}(p, j)\overline{α_{π}(p, k)})^{ν}}{νp^{νs}} \\ & = ∑_{ν=1}^{∞} \frac{|a_{π}(p^{ν})|^{2}}{νp^{νs}} \end{aligned}  (2.19)

With the local factors one can define the completed Rankin-Selberg L-function

Φ(s, π × π̃) = L(s, π_{∞} × π̃_{∞})L(s, π × π̃)  (2.20)

Some of the basic analytic properties of L(s, π × π̃) which we will use are as follows.

RS 1. [10] The Euler product for L(s, π × π̃) converges absolutely for Re s > 1 and L(s, π × π̃) has a simple pole at s = 1.

RS 2. Φ(s, π × π̃) has a meromorphic continuation to the entire complex plane and satisfies a functional equation

\begin{aligned}Φ(s, π × π̃) & = ϵ(s, π × π̃)Φ(1 − s, π × π̃) \\ ϵ(s, π × π̃) & = τ(π × π̃)Q_{π×π̃}^{−s} \end{aligned}

where Q_{π×π̃} > 0 and τ(π × π̃) = ±Q_{π×π̃}^{1/2}.

RS 3. Φ(s, π × π̃) is bounded in vertical strips, and is holomorphic except for simple poles at s = 0, 1.

There are two approaches to proving analytic properties of Φ(s, π × π̃). The first is via Rankin-Selberg integrals as developed by Jacquet, Piatetski-Shapiro and Shalika, and the second using the constant term of general Eisenstein series as is done by Shahidi and by Mœglin and Waldspurger. The first approach yields RS 1 and RS2 but the complicated nature of the archimedean integrals [11] makes RS 3 much more elusive by this method. On the other hand, the second method (which avoids such integrals) yields [19] that Φ(s, π × π̃) is entire except for simple poles at s = 0, 1. To see that s(1 − s)Φ(s, π × π̃) is of order one and bounded in vertical strips we can proceed as follows: As in the first part of [11] choose Whittaker functions W_{∞} and W′_{∞} for π_{∞} and π̃_{∞}, and ϕ ∈ S(R^{m}). The archimedean integrals Ψ(s, W_{∞}, W′_{∞}, ϕ) satisfy

g(s, W_{∞}, W′_{∞}, ϕ) := \frac{Ψ(s, W_{∞}, W′_{∞}, ϕ)}{L(s, π_{∞} × π̃_{∞})}

is entire and satisfies a functional equation

g(1 − s, W̃_{∞}, W̃′_{∞}, ϕ̂) = ϵ(s)g(s, W_{∞}, W′_{∞}, ϕ)

with ϵ(s) of the form ab^{s}. Moreover, note that Ψ(s, W_{∞}, W′_{∞}, ϕ) is bounded in vertical strips (except for a finite number of poles in the strip in question) and is uniformly bounded for Re s ≫ 0. It follows that g(s, W_{∞}, W′_{∞}, ϕ) is of order one. Now using the global Rankin-Selberg integral, one checks that

s(1 − s)Φ(s, π × π̃) = \frac{B(s, W_{∞}, W′_{∞}, ϕ)}{g(s, W_{∞}, W′_{∞}, ϕ)}

where B(s) is entire of order one (it comes from an integral against a standard Eisenstein series). On the other hand by [19] we know that s(1 − s)Φ(s, π × π̃) is entire, and so it must be of order one. Moreover Φ(s, π × π̃) is bounded for Re s ≫ 1 and hence by the functional

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 15

equation RS 2 this is also so for Re s ≪ 0. By an application of the Phragmen-Lindelöf principle the claim follows.

As an application of RS 1 we obtain the asymptotics of

σ(x) := ∑_{log n≤x} \frac{|c_{π}(n)|^{2}}{n}  (2.21)

We note that the bound (2.3) ensures that the contribution to σ(x) of n = p^{e} for ramified primes p ∈ S_{π} is bounded independently of x. As for the unramified primes, set L_{S}(s, π × π̃) = ∏_{p∉S} L(s, π_{p} × π̃_{p}) (sometime called the partial L-function). Differentiating (2.19) we see

\frac{L′_{S}}{L_{S}} (s, π × π̃) = − ∑_{(n,S_{π})=1} \frac{Λ(n)|a_{π}(n)|^{2}}{n^{s}}  (2.22)

Differentiating (2.22) we have

G(s) := (\frac{L′_{S}}{L_{S}} )′ (s + 1, π × π̃) = ∑_{(n,S_{π})=1} \frac{(log n)Λ(n)|a_{π}(n)|^{2}}{n^{s+1}}(2.23)

Since L(s, π × π̃) has a simple pole at s = 1, it follows from (2.23) that

∑_{n=1}^{∞} \frac{(log n)Λ(n)|a_{π}(n)|^{2}}{n} n^{−s} ∼ \frac{1}{s^{2}} as s ↘ 0 (s real)

Hence by a standard Tauberian argument we conclude that

σ_{1}(x) := ∑_{n≤x} \frac{(log n)Λ(n)|a_{π}(n)|^{2}}{n} ∼ \frac{log^{2} x}{2}  (2.24)

To relate σ_{1}(x) and σ(x) we need to make a technical hypothesis (this is the hypothesis (1.7)):

Hypothesis H. For any fixed k ≥ 2

∑_{p} \frac{|(log p)a_{π}(p^{k})|^{2}}{p^{k}} < ∞

We will establish H in many cases below. Note that it is an immediate consequence of the “Ramanujan conjectures” mentioned after (2.2). Indeed these assert that |a_{π}(p^{k})| ≤ m which implies H with lots to spare. In view of (2.3) we see that if k > (m^{2} + 1)/2 then ∑_{p} \frac{|(log p)a_{π}(p^{k})|^{2}}{p^{k}} < ∞. Hence assuming H we have

∑_{k≥2} ∑_{p} \frac{|(log p)a_{π}(p^{k})|^{2}}{p^{k}} < ∞  (2.25)

so that

σ_{1}(x) = σ(x) + O(1)  (2.26)

Proposition 2.2. Assuming H we have σ(x) ∼ (log x)^{2}/2.

That this asymptotic is independent of π is at the root of the universality of GUE. Using RS 3 we can sharpen Proposition 2.2 somewhat.

Proposition 2.3. Assuming H, we have

∑_{n≤x} \frac{|c_{π}(n)|^{2}}{n} = \frac{log^{2} x}{2} + O(log x)

Proof. Since the ramified primes contribute only a bounded quantity, we need only estimate the sum over (n, S_{π}) = 1. The function G(s) in (2.23) is holomorphic for Re s > 0, with Taylor expansion at s = 0 of the form 1/s^{2}+ holomorphic. G(s) is meromorphic and has at most double poles in Re s ≤ 0. A term-by-term integration yields the familiar identity

∑_{n≤x}(1 − \frac{n}{x})\frac{|(log n)Λ(n)a_{π}(n)|^{2}}{n} = \frac{1}{2πi} ∫_{Re s=1} G(s)\frac{x^{s}}{s(s + 1)}ds + O(1) (2.27)

(the O(1) term coming from the ramified primes and from (2.25)). Now RS 3 allows us to give standard bounds for G(s) in Re s < 0 and also to bound the number of poles of G(s) in |s| < T by O_{ϵ}(T^{1+ϵ}). In particular ∑ 1/|ρ|(|ρ| + 1) < ∞, where the sum is over the zeros of L(s, π × π̃). So shifting the contour in (2.27) to the left of Re s = 0 yields

∑_{n≤x}(1 − \frac{n}{x})\frac{|c_{π}(n)|^{2}}{n} = Res_{s=0} \frac{G(s)x^{s}}{s(s + 1)} + O(∑_{ρ≠0} \frac{log x}{|ρ|(|ρ| + 1)})

= \frac{log^{2} x}{2} + O(log x) (2.28)

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 17

If f(x) = σ(log x), σ as in (2.21) then

\frac{1}{x} ∫_{1}^{x} f(t)dt = ∑_{n≤x}(1 − \frac{n}{x})\frac{|c_{π}(n)|^{2}}{n} + O(1)

hence

∫_{1}^{x} f(t)dt = \frac{x log^{2} x}{2} + O(x log x)  (2.29)

Since f is increasing, we have for any h ≤ x:

\frac{1}{h} ∫_{x−h}^{x} f(t)dt ≤ f(x) ≤ \frac{1}{h} ∫_{x}^{x+h} f(t)dt

Applying (2.29) with h = x/4 yields Proposition 2.3.

We turn to the “technical hypothesis” H. There is little doubt about its truth, since as was pointed out it follows from very modest bounds towards the “Ramanujan conjectures” (which are proven for some of the known π’s on GL_{m}). Even so, we have not been able to establish it in general.

Proposition 2.4. Hypothesis H holds for 1 ≤ m ≤ 3.

Proof. For m = 1 this is trivial. We give the proof for m = 3. For m = 2 it is proven in the same way (or follows from known bounds in that case). Write A_{π}(p) = diag(α_{1}, α_{2}, α_{3}) so that a_{π}(p^{k}) = tr A_{π}(p)^{k}. Assume that |α_{1}| ≥ |α_{2}| ≥ |α_{3}|. Since ω_{π}(p) = det A_{π}(p) has absolute value 1 and \{\overline{α_{j}}\} = \{α_{j}^{−1}\} by (2.8) , we must have |α_{2}| = 1 and |α_{3}| = 1/|α_{1}|. Therefore

|α_{1}| + |α_{2}| + |α_{3}| ≤ |α_{1} + α_{2} + α_{3}| + 4

and

|a_{π}(p^{k})|^{2} ≪ 1 + |a_{π}(p)|^{2k}

Together with |a_{π}(p)| ≪ p^{1/2−1/10} (see (2.3)) we get

∑_{p} \frac{|(log p)^{2}|a_{π}(p^{k})|^{2}}{p^{k}} ≪ ∑_{p} \frac{log^{2} p}{p^{k}} + ∑_{p} \frac{log^{2} p|a_{π}(p)|^{2}}{p^{1+2(k−1)/5}}

Since k > 1 we can apply RS 1 (in particular the convergence in Re s > 1) to conclude that ∑_{p} \frac{|(log p)^{2}|a_{π}(p^{k})|^{2}}{p^{k}} < ∞.

18 ZEÉV RUDNICK AND PETER SARNAK

For the rest of the paper we will assume that either m ≤ 3 or that Hypothesis H is valid.

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 19

3. Sums over primes

We wish to study the asymptotic behaviour of the n-level correlation function

R_{n}(f, T) = \frac{1}{N} ∑_{i_{1},...,i_{n}≤N}^{∗} f(γ̃_{i_{1}}, …, γ̃_{i_{n}})  (3.1)

where N = N(T) and ∑^{∗} means we sum over distinct indices i_{j}. Instead of looking directly at R_{n}(f, T), we instead look at the sums

C_{n}(f, T) = ∑_{i_{1},...,i_{n}≤N} f(γ̃_{i_{1}}, …, γ̃_{i_{n}})  (3.2)

It is important to note that the sum in (3.2) is no longer over distinct ordered zeros as in the definition of the n-level correlation function (3.1). We will recover (3.1) from (3.2) by combinatorial sieving in §4. In order to determine the asymptotics of C_{n}(f, T) we look at smoothed sums

C_{n}(f, h, T) := ∑_{γ_{1},...,γ_{n}} h_{1}(\frac{γ_{1}}{T}) … h_{n}(\frac{γ_{n}}{T})f(\frac{L}{2π} γ_{1}, …, \frac{L}{2π} γ_{n}) (3.3)

where we have set

L = m log T  (3.4)

and h_{j}(r) is a smooth “cutoff” - we take

h_{j}(r) = ∫_{−∞}^{∞} g_{j}(u)e^{iru}du  (3.5)

with g_{j} ∈ C_{c}^{∞}(R). Our main result in this section is Theorem 3.1 which gives the asymptotics of C_{n}(f, h, T). We prove it for f satisfying TF 2,3, though in the end one is only interested in looking at symmetric f. Our reason for considering these more general f’s is to carry out the induction in Section 4. In fact, it will be convenient to work with the Fourier transform of f; thus for Φ a compactly supported C^{1} function on R^{n} we get an f satisfying TF 2 by setting

f(x) = ∫_{R^{n}} Φ(ξ)δ(ξ_{1} + ⋯ + ξ_{n})e(−x · ξ)dξ .  (3.6)

In the sequel, we set for h = (h_{1}, …, h_{n})

κ(h) = ∫_{−∞}^{∞} h_{1}(r) … h_{n}(r)dr  (3.7)

20 ZEÉV RUDNICK AND PETER SARNAK

Theorem 3.1. Let Φ ∈ C^{1}(R^{n}) be supported in ∑_{j=1}^{n} |ξ_{j}| < 2/m, and let f(x) = ∫_{R^{n}} Φ(ξ)δ(ξ_{1} + ⋯ + ξ_{n})e(−x · ξ)dξ. Then for h_{j} as in (3.5), we have

∑ h_{1}(\frac{γ_{1}}{T}) … h_{n}(\frac{γ_{n}}{T})f(\frac{L}{2π} γ_{1}, …, \frac{L}{2π} γ_{n}) = κ(h)\frac{TL}{2π} ∫_{R^{n}} C\underline{O}(v)Φ(v)dv + O(T)

(3.8)

with

∫_{R^{n}} Φ(v)C\underline{O}(v)dv = Φ(0) +

∑_{r=1}^{[n/2]} ∑ ∫ ⋯ ∫ |v_{1}| … |v_{r}|Φ(v_{1}e_{i(1),j(1)} + ⋯ + v_{r}e_{i(r),j(r)})dv_{1} … dv_{r} (3.9)

where the sum is over all choices of r disjoint pairs of indices i(t) < j(t) in {1, …, n} and for i < j we set

\begin{cases}e_{i,j} & = e_{i} − e_{j} \\ e_{i} & = (0, …, 1, 0, …) the i-th standard basis vector. \end{cases} (3.10)

From Theorem 3.1, we will deduce the asymptotics of the unsmoothed sums C_{n}(f, T); for this we will assume the Riemann Hypothesis for L(s, π):

Theorem 3.2. Let Φ ∈ C^{2}(R^{n}) be supported in ∑ |ξ_{j}| < 2/m, and f be given by (3.6). Assume the Riemann Hypothesis for L(s, π), then

C_{n}(f, T) ∼ N(T) ∫_{R^{n}} Φ(u)C\underline{O}(u)du + O(T)

Proof of Theorem 3.1: To begin the proof, we rewrite the sum C_{n}(f, h, T) using the Fourier transform as

C_{n}(f, h, T) = ∫_{R^{n}} ∏_{j=1}^{n} \{∑_{γ_{j}} h_{j}(\frac{γ_{j}}{T})e^{−iLγ_{j}ξ_{j}} \} Φ(ξ)δ(ξ_{1} + ⋯ + ξ_{n})dξ (3.11)

We can convert (3.11) into a sum over primes by use of the Explicit Formula (2.16), with the test functions

H_{T} (r) = h_{j}(\frac{r}{T})e^{−irLξ},  G_{T} (u) = T g_{j}(T (Lξ + u)) (3.12)

where g_{j}(u), h_{j}(r) are as in (3.5), and ξ ∈ R.

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 21

The Explicit Formula (2.16) with this choice reads

∑ h(\frac{γ}{T})e^{−iLγξ} = δ(π) \{h(−\frac{i}{2T})T^{−mξ/2} + h(\frac{i}{2T})T^{mξ/2}\}

+ log Q_{π} · T g(T Lξ)

+ \frac{1}{2π} ∫_{−∞}^{∞} ∑_{j=1}^{m} (\frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + μ_{π}(j) + ir) + \frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + \overline{μ_{π}(j)} − ir)) h(\frac{r}{T})e^{−irLξ}dr

− T ∑_{n=1}^{∞} \frac{Λ(n)}{\sqrt{n}} \{a(n)g(T (Lξ + log n)) + \overline{a(n)}g(T (Lξ − log n))\}

= polar + T g_{T} (T Lξ) + T S^{+}(ξ) + T S^{−}(ξ)

(3.13)

where

polar = δ(π) \{h(−\frac{i}{2T})T^{−mξ/2} + h(\frac{i}{2T})T^{mξ/2}\}  (3.14)
g_{T} (x) = \frac{1}{2π} ∫_{−∞}^{∞} h(r)Ω_{π}(rT)e^{−irx}dr  (3.15)

Ω_{π}(r) = log Q_{π} + ∑_{j=1}^{m} \frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + μ_{π}(j) + ir) + \frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + \overline{μ_{π}(j)} − ir)

(3.16)

S^{+}(ξ) = − ∑_{n=1}^{∞} \frac{Λ(n)a(n)}{\sqrt{n}}g(T (Lξ + log n))
S^{−}(ξ) = − ∑_{n=1}^{∞} \frac{Λ(n)\overline{a(n)}}{\sqrt{n}}g(T (Lξ − log n))

(3.17)

(the term polar occurs only in the case of ζ(s); in the sequel we omit it).

Inserting (3.13) into (3.11) with the different h_{j}, we find that we have expressed C_{n}(f, h, T) as a sum over primes as desired:

C_{n}(f, h, T) = ∫_{R^{n}} ∏_{j=1}^{n} {T g_{j,T} (T Lξ_{j}) + T S_{j}^{+}(ξ_{j}) + T S_{j}^{−}(ξ_{j})} Φ(ξ)δ(ξ_{1} + ⋯ + ξ_{n})dξ (3.18)

Expanding the product in (3.18), we find that C_{n}(f, h, T) is an alternating sum of terms of the form

C_{r,s}(T) = ∑ \frac{c(n_{1}) … c(n_{r})\overline{c(n_{r+1})} … \overline{c(n_{r+s})}}{\sqrt{n_{1} · · · · · n_{r+s}}}A_{r,s}(n, T) (3.19)

where we have set

c(n) = Λ(n)a(n)  (3.20)

22 ZEÉV RUDNICK AND PETER SARNAK

and

A_{r,s}(n, T) = T^{n} ∫ ∏_{j=1}^{r} g_{j}(T (Lξ_{j} + log n_{j})) ∏_{j=r+1}^{r+s} g_{j}(T (Lξ_{j} − log n_{j}))

· ∏_{j>r+s} g_{j,T} (T Lξ_{j}) · Φ(ξ)δ(ξ_{1} + ⋯ + ξ_{n})dξ (3.21)

In expressing C_{n}(f, h, T) as a sum of various C_{r,s}(T) we get terms From all possible choices of r of the factors to be S_{j}^{+}(ξ_{j}), s of the factors to be S_{j}^{−}(ξ_{j}), and the remaining k = n−r −s of the factors to be g_{j,T} (T Lξ_{j}).

Lemma 3.1. 1.

g_{T} (x) ≪ \begin{cases}log T, & |x| ≪ log log T \\ \frac{1}{|x|}, & |x| ≫ log log T \end{cases}

2. ∫ |g_{T} (x)|dx ≪ log T.

Proof. Recall that by (3.16),

g_{T} (x) = \frac{1}{2πi} ∫_{Re s=0} h(\frac{s}{i})ω_{π}(T s)e^{−sx}ds

with

ω_{π}(s) = log Q_{π} + ∑_{j=1}^{m} (\frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + μ_{π}(j) + s) + \frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + \overline{μ_{π}(j)} − s))

(3.22)

Assuming that x > 0, we shift the contour of integration to the right to s = σ + ir, with σ > 0; this we can do since by Stirling’s formula, Ω_{π}(r) ≪ log(r) and h(r) is rapidly decreasing as | Re(r)| → ∞. Since Re(1/2+μ_{π}(j)) > 0 (2.5), the first Γ-factor is holomorphic in Re s ≥ 0. The second factors contribute simple poles at

s_{k}^{(j)} = \frac{1/2 + μ̄_{π}(j) + 2k}{T} , 0 ≤ k ≤ \frac{σ}{2} T

Thus

g_{T} (x) = \frac{1}{2πi} ∫_{Re s=σ} h(\frac{s}{i})ω_{π}(sT)e^{−sx}ds + \frac{1}{T} ∑_{j=1}^{m} ∑_{0≤k≤ \frac{σ}{2} T} h(\frac{s_{k}^{(j)}}{i})e^{−s_{k}^{(j)}x}

The double sum is majorized by

\frac{1}{T} ∑_{j=1}^{m} ∑_{0≤k≤ \frac{σ}{2} T} e^{−(1/2+μ̄_{π}(j)+2k)x/T} ≪ \frac{e^{−αx/T}}{x} (1 − e^{−σx/2})

where α = min_{1≤j≤m}{1/2 + Re μ_{π}(j)} > 0.

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 23

As for the integral, by Stirling’s formula

\frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + μ̄_{π}(j) ± sT) ≪ log((1 + |s|)T)

and since h(\frac{s}{i}) is rapidly decreasing in vertical strips, we can bound the integral by

\frac{1}{2πi} ∫_{Re s=σ} h(\frac{s}{i})ω_{π}(sT)e^{−sx}ds ≪ e^{−σx} log T

Thus we see that

g_{T} (x) ≪ \frac{e^{−αx/T}}{x} (1 − e^{−σx/2}) + e^{−σx} log T

giving part (1). Part (2) follows from integrating this.

From Lemma 3.1 we see that the integrals defining A_{r,s}(n, T) are rapidly convergent.

Lemma 3.2. Let Φ as in (3.6) be supported in |ξ_{1}| + ⋯ + |ξ_{n}| ≤ \frac{2−δ}{m}. Then A_{r,s}(n, T) = 0 unless |n_{j}| ≪ T and n_{1}n_{2} · · · · · n_{r+s} ≪ T^{2−δ}.

Proof. The integrand in (3.21) is zero unless there is an η ∈ Supp Φ (so ∑_{j} η_{j} = 0) such that

\begin{cases}|T (η_{j}L + log n_{j})| ≪ 1, & j = 1, …, r \\ |T (η_{j}L − log n_{j})| ≪ 1, & j = r + 1, …, r + s \end{cases}  (3.23)

Hence

\frac{n_{j}}{T^{m|η_{j}|}} = 1 + O(\frac{1}{T})  (3.24)

so that n_{j} ≪ T^{m|η_{j}|} ≪ T^{1−δ/2} and n_{1}n_{2} · · · · · n_{r+s} ≪ T^{m ∑ |η_{j}|} ≪ T^{2−δ}.

What follows are a series of reductions which show that the main term in C_{r,s}(T) comes from the “diagonal sums”.

Lemma 3.3. Let Ã_{r,s}(n, T) be as in (3.21) with the region of integration in the variables ξ_{j}, j > r+s, restricted to |T Lξ_{j}| ≪ T^{δ/3}. Then for T sufficiently large, Ã_{r,s}(n, T) = 0 unless n = (n_{1}, …, n_{r+s}) satisfies the conclusion of Lemma 3.2 and in addition

n_{1} … n_{r} = n_{r+1} … n_{r+s}  (3.25)

Proof. Since g has compact support, in order that the integrand not vanish we need some η ∈ Supp Φ, that is

\begin{cases}|T (η_{j}L + log n_{j})| ≪ 1, & j = 1, …, r \\ |T (η_{j}L − log n_{j})| ≪ 1, & j = r + 1, …, r + s \\ |T Lη_{j}| ≪ T^{δ/3}, & j > r + s . \end{cases}  (3.26)

Furthermore, ∑_{j=1}^{n} η_{j} = 0, hence

| log \frac{n_{1} … n_{r}}{n_{r+1} … n_{r+s}} | = | ∑_{j=1}^{r}(Lη_{j} + log n_{j}) + ∑_{j=r+1}^{r+s}(Lη_{j} − log n_{j}) + ∑_{j>r+s} Lη_{j}| ≪ \frac{1}{T} + T^{δ/3−1}

(3.27)

Thus

| log \frac{n_{1} … n_{r}}{n_{r+1} … n_{r+s}} | ≪ T^{−1+δ/3} .  (3.28)

Setting M = n_{1} … n_{r}, N = n_{r+1} … n_{r+s}, we know that MN ≪ T^{2−δ} and that | log M/N| ≪ T^{−1+δ/3}. Assume that M ≠ N, say M = N +u, u ≥ 1. Then

T^{−1+δ/3} ≫ | log \frac{M}{N} | = log(1 + \frac{u}{N}) ≥ \frac{1}{N} ≥ \frac{1}{\sqrt{MN}} ≫ T^{−1+δ/2}(3.29)

Since δ > 0, this gives a contradiction for T large. Thus M = N establishing the Lemma.

Recall the sum C_{r,s}(T) given in (3.19) and denote by C̃_{r,s}(T) the corresponding sum with A_{r,s}(n, T) replaced by Ã_{r,s}(n, T).

Proposition 3.1. C_{r,s}(T) = C̃_{r,s}(T) + O(T^{1−δ/3})

We begin by estimating the difference between A_{r,s}(n, T) and Ã_{r,s}(n, T). As in the proof of Lemma 3.3, we set M = n_{1} … n_{r}, N = n_{r+1} … n_{r+s}.

Lemma 3.4. If NM ≪ T^{2−δ} then

A_{r,s}(n, T) − Ã_{r,s}(n, T) ≪ \begin{cases}T^{1−δ/3}L^{−r−s}, & | log M/N| ≪ T^{δ/3−1} \\ \frac{L^{−r−s}}{| log M/N|}, & | log M/N| ≫ T^{δ/3−1} \end{cases}

Proof. The region of integration for the difference A_{r,s}(n, T)−Ã_{r,s}(n, T) is a union U of the sets F_{∥} = {ξ : |T Lξ_{∥}| ≫ T^{δ/∋}}, k > r + s. Without

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 25

loss of generality we estimate this integral over the region F_{\}. For this purpose, set

x_{j} = \begin{cases}T (Lξ_{j} + log n_{j}), & 1 ≤ j ≤ r \\ T (Lξ_{j} − log n_{j}), & r + 1 ≤ j ≤ r + s \\ T Lξ_{j}, & j > r + s \end{cases}

We have on changing variables

A_{r,s}(n, T) − Ã_{r,s}(n, T) = T^{n} ∫_{U} ∏_{j=1}^{r} g_{j}(T (Lξ_{j} + log n_{j})) ∏_{j=r+1}^{r+s} g_{j}(T (Lξ_{j} − log n_{j}))

· ∏_{j>r+s} g_{j,T} (T Lξ_{j}) · Φ(ξ)δ(ξ_{1} + ⋯ + ξ_{n})dξ

≪ \frac{T}{L^{n−1}} ∫_{\{|x_{j}|≪T L, |x_{n}|≫T^{δ/3}\}} ∏_{j≤r+s} |g_{j}(x_{j})| ∏_{r+s<j≤n−1} |g_{j,T} (x_{j})|

· |g_{n,T} (T log M − T log N − ∑_{j=1}^{n−1} x_{j})|dx_{1} … dx_{n−1}

We claim that

∫_{{|x_{j}|≪T L, |x_{n}|≫T^{δ/3}}} | ∏_{j≤r+s} g_{j}(x_{j}) ∏_{r+s<j≤n−1} g_{j,T} (x_{j})g_{n,T} (T log M/N − ∑_{j=1}^{n−1} x_{j})|dx_{1} … dx_{n−1}

≪ \begin{cases}\frac{L^{n−1−r−s}}{T^{δ/3}} , & | log M/N| ≪ T^{δ/3−1} \\ \frac{L^{n−1−r−s}}{T | log M/N|}, & | log M/N| ≫ T^{δ/3−1} \end{cases} (3.30)

To see this, first assume that | log M − log N| ≪ T^{δ/3−1}. As in the proof of Lemma 3.3, since MN ≪ T^{2−δ}, this implies that M = N and so in this case on using Lemma 3.1 the integral in (3.30) is bounded by

∫_{|x_{j}|≪T L} | ∏_{j≤r+s} g_{j}(x_{j}) ∏_{r+s<j≤n−1} g_{j,T} (x_{j})|T^{−δ/3}dx_{1} … dx_{n−1} ≪ \frac{L^{n−1−r−s}}{T^{δ/3}}

If | log M −log N| ≫ T^{δ/3−1}, then we write the integral as a sum I_{1} +I_{2} of integrals over regions | ∑_{j=1}^{n−1} x_{j}| < \frac{1}{2}T | log M/N| and | ∑_{j=1}^{n−1} x_{j}| ≥ \frac{1}{2}T | log M/N|. In the first case, we have |x_{n}| = |T log M/N−∑_{j=1}^{n−1} x_{j}| ≫ T | log M/N| and then

I_{1} ≪ \frac{1}{|T log M − T log N|} ∫_{| ∑_{j=1}^{n−1} x_{j}|< \frac{1}{2} T | log M/N|} | ∏_{j≤r+s} g_{j}(x_{j}) ∏_{r+s<j≤n−1} g_{j,T} (x_{j})|dx

≪ \frac{L^{n−1−r−s}}{|T log M − T log N|}

26 ZEÉV RUDNICK AND PETER SARNAK

For the integral I_{2} over the region where | ∑ x_{j}| ≫ T | log M/N|, we write the region as a union of domains where for some j > r + s we have |x_{j}| ≫ T | log M/N| (this is possible since |x_{j}| ≪ 1 if j ≤ r + s). On such a domain we use

g_{T} (x_{j}) ≪ \frac{1}{|x_{j}|} ≪ \frac{1}{T | log M/N|}

(Lemma 3.1) to see that the integral I_{2} over say |x_{n−1}| ≫ T | log M/N| is bounded by

≪ \frac{1}{T | log M/N|} ∫_{\{x∈F_{\}:|§_{\−∞}|≫T | log M/N |\}} | ∏_{j=1}^{r+s} g_{j}(x_{j}) ∏_{j>r+s}^{n−2} g_{j,T} (x_{j})

·g_{n,T} (T log M/N − ∑_{j=1}^{n−1} x_{j})|dx_{1} … dx_{n−1}

Now set y = T log M/N − ∑_{j=1}^{n−1} x_{j} and change variable in the integral over x_{n−1}; in the region of integration it is bounded by

∫_{T^{δ/3}≪y≪T L} \frac{dy}{y} ≪ L

We find that

\frac{1}{T | log M/N|} ∫_{\{x∈F_{\}:|§_{\−∞}|≫T | log M/N |\}} ≪ \frac{L^{n−1−r−s}}{|T log M − T log N|}

This proves our claim (3.30) and so Lemma 3.4.

To prove Proposition 3.1 we divide the difference C_{r,s}(T) − C̃_{r,s}(T) into two sums Σ_{diag}+Σ_{off}, the first sum Σ_{diag} over n such that | log M/N| ≪ T^{δ/3−1}, which implies as before that M = N, and the second sum Σ_{off} over n for which T | log M/N| ≫ T^{δ/3}.

For the diagonal sum Σ_{diag}, we need to note that

∑_{\begin{matrix}M=N \\ n_{1}...n_{r+s}≪T^{2−δ} \end{matrix}} ∏_{j=1}^{r+s} \frac{|c(n_{j})|}{\sqrt{n_{j}}} ≪ (log T)^{r+s}

- we defer the proof to Lemma 3.9 where we will see a more precise result. Combining this with Lemma 3.4, we see that

Σ_{diag} ≪ T^{1−δ/3}  (3.31)

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 27

Next we handle the off-diagonal sum Σ_{off}. We have

Σ_{off} ≪ \frac{1}{L^{r+s}} ∑_{\begin{matrix}MN≪T^{2−δ} \\ M≠N \end{matrix}} \frac{1}{| log M/N|} ∏_{j=1}^{r+s} \frac{|c(n_{j})|}{\sqrt{n_{j}}}

Setting

a_{k}(M) := ∑_{n_{1}...n_{k}=M} ∏_{j=1}^{k} |c(n_{j})|

we have

Σ_{off} ≪ \frac{1}{L^{r+s}} ∑_{\begin{matrix}MN≪T^{2−δ} \\ M≠N \end{matrix}} \frac{a_{r}(M)a_{s}(N)}{\sqrt{MN}| log M/N|}  (3.32)

Lemma 3.5. For k ≥ 1 fixed, and any ϵ > 0

∑_{m≤X} a_{k}(m)^{2} ≪_{ϵ} X^{1+ϵ}

Proof. We begin by noting that on using Cauchy-Schwartz and the fact that the number of ways of writing m = m_{1} · · · · · m_{k} is O(m^{ϵ}) for any ϵ > 0, we have

a_{k}(m)^{2} ≪ m^{ϵ} ∑_{m_{1}·····m_{k}=m} |c(m_{1})|^{2} … |c(m_{k})|^{2}

and so

∑_{m≤X} a_{k}(m)^{2} ≪ X^{ϵ} ∑_{m_{1}·····m_{k}≤X} |c(m_{1})|^{2} … |c(m_{k})|^{2}  (3.33)

To estimate the above sum, we first note that

∑_{n≤X} |c(n)|^{2} ≪ X^{1+ϵ}  (3.34)

for all ϵ > 0, which follows immediately from the absolute convergence of (2.22) in Re(s) > 1 together with that series having non-negative coefficients. Next, we make a dyadic decomposition of the sum (3.33) into O((log X)^{k}) terms of the form

∑_{M_{1}≤m_{1}≤2M_{1}} ⋯ ∑_{M_{k}≤m_{k}≤2M_{k}} |c(m_{1})|^{2} … |c(m_{k})|^{2}

If M_{1} … M_{k} ≪ X then on using (3.34) we find

∑_{M_{1}≤m_{1}≤2M_{1}} ⋯ ∑_{M_{k}≤m_{k}≤2M_{k}} |c(m_{1})|^{2} … |c(m_{k})|^{2} ≪ M_{1}^{1+ϵ} … M_{k}^{1+ϵ} ≪ X^{1+ϵ}

which gives the desired estimate.

Returning to (3.32), we may without loss of generality assume that N < M, which implies that N ≪ T^{1−δ/2}. We consider two ranges in (3.32): the sum I_{∞} with N < M < 2N and the sum I_{∈} with N ≤ M/2. For the first case,

I_{∞} = ∑_{N≤T^{1−δ/2}} ∑_{N<M<2N} \frac{a_{r}(N)a_{s}(M)}{\sqrt{NM} log M/N}

≪ ∑_{N≤T^{1−δ/2}} ∑_{k=1}^{N} \frac{a_{r}(N)a_{s}(N + k)}{\sqrt{N(N + k)} log(1 + k/N)}

≪ ∑_{N≤T^{1−δ/2}} ∑_{k=1}^{N} \frac{a_{r}(N)a_{s}(N + k)}{k}

≪ ∑_{k≪T^{1−δ/2}} \frac{1}{k} ∑_{N≤2T^{1−δ/2}} (a_{r}(N)^{2} + a_{s}(N)^{2})

≪ (T^{1−δ/2})^{1+ϵ_{1}} log T

so that for all ϵ > 0

I_{∞} ≪ T^{∞−δ/∈+ϵ}  (3.35)

For the sum I_{∈}, when 2N < M then log M/N > log 2 so that

I_{∈} = ∑_{N ≤T^{∞−δ/∈}} ∑_{\begin{matrix}2N<M \\ MN<T^{2−δ} \end{matrix}} \frac{⊣_{∇}(N)⊣_{∫}(M)}{\sqrt{MN} log M/N} ≪ ∑_{MN <T^{∈−δ}} \frac{⊣_{∇}(N)⊣_{∫}(M)}{\sqrt{MN}}

Again by use of a dyadic decomposition, we will know

I_{∈} ≪ T^{∞−δ/∈+ϵ}  (3.36)

once we can show that for AB < T^{2−δ}

∑_{\begin{matrix}A≤N≤2A \\ B≤M≤2B \end{matrix}} \frac{a_{r}(N)a_{s}(M)}{\sqrt{MN}} ≪ (AB)^{1/2+ϵ}  (3.37)

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 29

Now the L.H.S. in (3.37) is

( ∑_{A≤N≤2A} \frac{a_{r}(N)}{\sqrt{N}}) ( ∑_{B≤M≤2B} \frac{a_{s}(M)}{\sqrt{M}})

≪ ( ∑_{A≤N≤2A} \frac{1}{N} )^{1/2} ( ∑_{A≤N≤2A} a_{r}(N)^{2})^{1/2} ( ∑_{B≤M≤2B} \frac{1}{M} )^{1/2} ( ∑_{B≤M≤2B} a_{s}(M)^{2})^{1/2}

≪ A^{1/2+ϵ_{1}}B^{1/2+ϵ_{1}}

by Lemma 3.5.

Combining the estimates (3.35) and (3.36) we conclude that

Σ_{off} ≪ T^{1−δ/2+ϵ}  (3.38)

for any ϵ > 0. From (3.31) and (3.38) we get Proposition 3.1.

We have seen that for T ≫ 1,

C_{r,s}(T) = ∑_{\begin{matrix}n_{j}≪T \\ n_{1}...n_{r}=n_{r+1}...n_{r+s} \\ n_{1}...n_{r+s}≪T^{2} \end{matrix}} ∏_{j=1}^{r} \frac{c(n_{j})}{n_{j}} ∏_{j=r+1}^{r+s} \overline{c(n_{j})}Ã_{r,s}(n, T) + O(T^{1−δ/3}) (3.39)

Now change variables in the integral (3.21) for Ã_{r,s}(n, T) (when n_{1} … n_{r} = n_{r+1} … n_{r+s}) by setting

y_{j} = \begin{cases}T (Lξ_{j} + log n_{j}), & 1 ≤ j ≤ r \\ T (Lξ_{j} − log n_{j}), & r + 1 ≤ j ≤ r + s \\ T Lξ_{j}, & j > r + s \end{cases}  (3.40)

Note that we still have ∑_{j} y_{j} = 0 and the region of integration is

V = \begin{cases}∑_{j} y_{j} = 0 \\ |y_{j}| ≪ 1, & j ≤ r + s \\ |y_{j}| ≪ T^{δ/3}, & j > r + s \end{cases}  (3.41)

We then get

Ã_{r,s}(n, T) = \frac{T}{L^{n−1}} ∫_{V} ∏_{j=1}^{r+s} g_{j}(y_{j}) ∏_{j>r+s} g_{j,T} (y_{j})

· Φ(\frac{y_{1}}{T L} − \frac{log n_{1}}{L} , …, \frac{y_{r+s}}{T L} + \frac{log n_{r+s}}{L} , …, \frac{y_{n}}{T L})dy (3.42)

Expanding Φ in a Taylor series about the point (−\frac{log n_{1}}{L} , …, \frac{log n_{r+s}}{L} , …, 0) (recall that Φ is C^{1}) and using the constraints (3.41) we see that

Ã_{r,s}(n, T) = \frac{T}{L^{n−1}} ∫_{V} ∏_{j=1}^{r+s} g_{j}(y_{j}) ∏_{j>r+s} g_{j,T} (y_{j})dy

· \{Φ(−\frac{log n_{1}}{L} , …, \frac{log n_{r+s}}{L} , …, 0) + O(T^{−1+δ/3})\} (3.43)

with the error term uniform in n subject to n_{j} ≪ T.

Lemma 3.6. Setting k = n − r − s, we have

∫_{V} ∏_{j=1}^{r+s} g_{j}(y_{j}) ∏_{j>r+s} g_{j,T} (y_{j})dy = \frac{1}{2π} κ(h)L^{k} + O(L^{k−1}) (3.44)

Proof. First we claim that

∫_{V} ∏_{j=1}^{r+s} g_{j}(y_{j}) ∏_{j>r+s} g_{j,T} (y_{j})dy = ∫_{∑ y_{j}=0} ∏_{j=1}^{r+s} g_{j}(y_{j}) ∏_{j>r+s} g_{j,T} (y_{j})dy + O(\frac{L^{n−1−r−s}}{T^{δ/3}} ) (3.45)

Indeed, the difference between the two integrals in (3.45) is an integral over the union ∪_{k>r+s}V_{k}, where

V_{k} = {∑_{j=1}^{n} y_{j} = 0 : |y_{k}| ≫ T^{δ/3}}

It suffices to estimate the integral in (3.45) over such V_{k}, say k = n:

\begin{aligned}S_{n} & := ∫_{V_{n}} | ∏_{j=1}^{r+s} g_{j}(y_{j}) ∏_{j>r+s} g_{j,T} (y_{j})|dy \\ & = ^{∫∫} _{|y_{1}+...y_{n−1}|≫T^{δ/3}} | ∏_{j=1}^{r+s} g_{j}(y_{j}) ∏_{j=r+s+1}^{n−1} g_{j,T} (y_{j})| · |g_{n,T} (− ∑_{j=1}^{n−1} y_{j})|dy_{1} … dy_{n−1} \end{aligned}

By Lemma 3.1 part (1), on V_{n}

|g_{n,T} (− ∑_{j=1}^{n−1} y_{j})| ≪ \frac{1}{| ∑_{j=1}^{n−1} y_{j}|} ≪ \frac{1}{T^{δ/3}}

so that

S_{n} ≪ \frac{1}{T^{δ/3}} ^{∫∫} | ∏_{j=1}^{r+s} g_{j}(y_{j}) ∏_{j=r+s+1}^{n−1} g_{j,T} (y_{j})|dy_{1} … dy_{n−1} ≪ \frac{L^{n−1−r−s}}{T^{δ/3}}

Applying Parseval’s equality to (3.45) gives

∫_{∑_{j} y_{j}=0} g_{1}(y_{1}) … g_{n}(y_{n})dy = \frac{1}{2π} ∫_{−∞}^{∞} h_{1}(r) … h_{n}(r)dr = \frac{1}{2π} κ(h) (3.46)

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 31

From which it follows that

∫_{∑ y_{j}=0} ∏_{j=1}^{r+s} g_{j}(y_{j}) ∏_{j>r+s} g_{j,T} (y_{j})dy = \frac{1}{2π} ∫_{−∞}^{∞} h_{1}(r) … h_{n}(r)Ω_{π}(T r)^{k}dr (3.47)

On using Stirling’s approximation, we get for |r| ≥ 1 that Ω_{π}(T r) = m log(T r) + O(1) so that

\frac{1}{2π} ∫_{−∞}^{∞} h_{1}(r) … h_{n}(r)Ω_{π}(T r)^{k}dr = \frac{1}{2π} κ(h)L^{k} + O(L^{k−1}) (3.48)

and the lemma follows.

We can conclude from the above:

Lemma 3.7. For r + s > 0,

C_{r,s}(T) = \frac{1}{2π} κ(h)\frac{T}{L^{r+s−1}} ∑_{\begin{matrix}n_{j}≪T \\ n_{1}...n_{r}=n_{r+1}...n_{r+s} \\ n_{1}...n_{r+s}≪T^{2} \end{matrix}} ∏_{j=1}^{r} \frac{c(n_{j})}{\sqrt{n_{j}}} ∏_{j=r+1}^{r+s} \frac{\overline{c(n_{j})}}{\sqrt{n_{j}}}

· Φ(−\frac{log n_{1}}{L} , …, \frac{log n_{r+s}}{L} , …, 0) + O(T)

(3.49)

Proof. From (3.43) and (3.44), we see that for n in the range of summation,

Ã_{r,s}(n, T) = \frac{T}{L^{r+s−1}} \frac{1}{2π} κ(h)Φ(−\frac{log n_{1}}{L} , …, \frac{log n_{r+s}}{L} , …, 0) + O(\frac{T}{L^{r+s}} )

(3.50)

and so

C_{r,s}(T) = \frac{1}{2π} κ(h)\frac{T}{L^{r+s−1}} \{1 + O(\frac{1}{L})\}

· ∑_{\begin{matrix}n_{j}≪T \\ n_{1}...n_{r}=n_{r+1}...n_{r+s} \\ n_{1}...n_{r+s}≪T^{2} \end{matrix}} ∏_{j=1}^{r} \frac{c(n_{j})}{n_{j}} ∏_{j=r+1}^{r+s} \overline{c(n_{j})} · Φ(−\frac{log n_{1}}{L} , …, \frac{log n_{r+s}}{L} , …, 0) (3.51)

We will see below (Lemma 3.9) that the sum is O(L^{r+s}), and so once that is established Lemma 3.7 will follow.

Lemma 3.8. Assume r + s ≥ 3. Then

∑_{p} ∑_{l_{1}+···+l_{s}=k_{1}+···+k_{r}} |\frac{c(p^{k_{1}}) · · · · · c(p^{k_{r}})\overline{c(p^{l_{1}})} · · · · · \overline{c(p^{l_{s}})}}{p^{k_{1}+...k_{r}}}| < ∞(3.52)

(the sums with k_{i}, l_{j} ≥ 1).

Proof. We first omit the restriction ∑ k_{i} = ∑ l_{j}, set t = r + s ≥ 3, and write k_{r+1} = l_{1}, …, k_{t} = l_{s}. The sum (3.52) is bounded by the sum

∑_{p} ∑_{k_{1},...,k_{t}≥1} \frac{|c(p^{k_{1}}) … c(p^{k_{t}})|}{p^{(k_{1}+···+k_{t})/2}}  (3.53)

Recall (2.3) which asserts that a(p^{k}) ≪ p^{k(1/2−β)} for β = 1/(m^{2}+1) > 0. We use this to bound the sum (3.53) when we restrict the exponents to k_{1} + ⋯ + k_{t} ≥ K > 1/β:

\begin{matrix}∑_{p} ∑_{k_{1}+···+k_{t}≥K} \frac{|c(p^{k_{1}}) … c(p^{k_{t}})|}{p^{(k_{1}+···+k_{t})/2}} & ≪ ∑_{p} ∑_{k≥K} \frac{p^{k(1/2−β)} log^{t} p}{p^{k/2}}k^{t} \\ ≪ ∑_{p} \frac{log^{t} p}{p^{Kβ}} < ∞ \end{matrix} (3.54)

To deal with the sum ∑ k_{j} < K, we need to use the Rankin-Selberg L-function. It suffices to show that for s fixed, s < K, the sum

∑_{p} ∑_{k_{1}+...k_{t}=s} \frac{|c(p^{k_{1}}) … c(p^{k_{t}})|}{p^{s/2}}  (3.55)

is bounded.

Recall that t ≥ 3 and so we may replace all but two of the coefficients c(p^{k}) by p^{k(1/2−β)} log p. Doing this we find that for fixed p, the summand in (3.55) is bounded by

\begin{matrix}∑_{k_{1}+k_{2}≤s−(t−2)} \frac{|c(p^{k_{1}})c(p^{k_{2}})|}{p^{(k_{1}+k_{2})/2}} ∑_{k_{3}+···+k_{t}=s−k_{1}−k_{2}} \frac{log^{t−2} p}{p^{(k_{3}+...k_{t})β}} & ≪_{s} ∑_{k_{1}+k_{2}≤s−(t−2)} \frac{|c(p^{k_{1}})c(p^{k_{2}})|}{p^{(k_{1}+k_{2})/2}} \frac{log^{t−2} p}{p^{(t−2)β}} \\ ≪ ∑_{1≤k_{1},k_{2}≤s} \frac{|c(p^{k_{1}})|}{p^{(k_{1}+δ′)/2}} \frac{|c(p^{k_{2}})|}{p^{(k_{2}+δ′)/2}} \end{matrix} (3.56)

for some δ′ > 0. Now fix k_{1}, k_{2} ≤ s as we may, sum over p and use Cauchy-Schwarz:

∑_{p} \frac{|c(p^{k_{1}})|}{p^{(k_{1}+δ′)/2}} \frac{|c(p^{k_{2}})|}{p^{(k_{2}+δ′)/2}} ≪ \{∑_{p} \frac{|c(p^{k_{1}})|^{2}}{p^{k_{1}+δ′}} \}^{1/2} \{∑_{p} \frac{|c(p^{k_{2}})|^{2}}{p^{k_{2}+δ′}} \}^{1/2}(3.57)

Each of the sums is now seen to converge by applying RS 1.

We use Lemma 3.8 to deal with sums over several prime factors.

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 33

Lemma 3.9. If 1 ≤ r ≤ s then

∑_{\begin{matrix}p_{1}^{k_{1}}...p_{r}^{kr} =q_{1}^{l_{1}}...q_{s}^{ls} \\ p_{i}^{k_{i}},q_{j}^{l_{j}} ≤x \end{matrix}} \frac{c(p_{1}^{k_{1}}) · · · · · c(p_{r}^{k_{r}})\overline{c(q_{1}^{l_{1}})} · · · · · \overline{c(q_{s}^{l_{s}})}}{p_{1}^{k_{1}} … p_{r}^{k_{r}}} = \begin{cases}O((log x)^{2r}), & r = s \\ O((log x)^{2r−2}), & r < s \end{cases} (3.58)

Proof. We divide the sum (3.58) into sub-sums according to the number of distinct prime factors appearing, and we collect together factors corresponding to the same prime. The sum (3.58) then becomes a sum of products of the form

∏_{p_{i}≠p_{j}} ∑_{p_{j}^{k_{1}},...,p_{j}^{ka}≤x} _{∑ k_{i}=∑ l_{j}}^{∑} \frac{c(p_{j}^{k_{1}}) · · · · · c(p_{j}^{k_{a}})\overline{c(p_{j}^{l_{1}})} · · · · · \overline{c(p_{j}^{l_{b}})}}{p_{j}^{k_{1}+···+k_{a}}}(3.59)

where the product is taken over distinct primes. There are at most r factors, and by Lemma 3.8, each factor in (3.59) contributes a bounded quantity unless a = b = 1, in which case it is bounded by O(log^{2} x). Thus the product (3.59) is O(log^{2r−2} x) unless r = s and a = b = 1 in each factor of (3.59). Thus if r < s then (3.58) is O(log^{2r−2} x), while for r = s we can relax the condition that the product in (3.59) is over distinct primes at the cost of introducing a bounded error in each factor, which after multiplying will introduce an error of O(log^{2r−2} x). Therefore we find on using Proposition 2.3 that for r = s, (3.58) equals

∏_{j=1}^{r} ∑_{p_{j}^{k_{j}} ≤x} \frac{|c(p_{j}^{k_{j}})|^{2}}{p_{j}^{k_{j}}} + O(log^{2r−2} x) = \frac{log^{2r} x}{2^{r}} + O(log^{2r−1} x) . (3.60)

Recall from Lemma 3.7 that

C_{r,s}(T) = \frac{1}{2π} κ(h)\frac{T}{(L)^{r+s−1}} ·

∑_{\begin{matrix}n_{1}...n_{r}=n_{r+1}...n_{r+s} \\ n_{1}...n_{r+s}≪T^{2} \end{matrix}} \frac{c(n_{1}) … c(n_{r})\overline{c(n_{r+1})} … \overline{c(n_{r+s})}}{n_{1} … n_{r}}Φ(−\frac{log n_{1}}{L} , …, \frac{log n_{r+s}}{L} , 0, …, 0)

+ O(T)

(3.61)

Lemma 3.10. 1. C_{r,s}(T) = O(T) unless r = s > 0.

2. If r = s then

C_{r,r}(T) = \frac{T L}{2π}κ(h) · ∑_{σ∈S_{r}} ∫_{0}^{1/m}⋯ ∫_{0}^{1/m} v_{1} … v_{r}

· Φ(−v_{1}, …, −v_{r}, v_{σ(1)}, …, v_{σ(r)}, 0, …, 0)dv_{1} … dv_{r} + O(T)

where S_{r} is the permutation group on r letters.

Proof. The first statement follows from the upper bound in Lemma 3.9. As for the second, we use summation by parts and the asymptotics in Proposition 2.3: From Lemma 3.8, we know that only distinct primes (not higher powers) are going to contribute to the main term. Thus

C_{r,r}(T) = \frac{1}{2π} κ(h)\frac{T}{(L)^{2r−1}} · ∑_{p_{1},...,p_{r}≤T} ∏_{j=1}^{r} \frac{|c(p_{j})|^{2}}{p_{j}}

· ∑_{q_{1}·····q_{r}=p_{1}·····p_{r}} Φ(−\frac{log p_{1}}{L} , …, −\frac{log p_{r}}{L} , \frac{log q_{1}}{L} , …, \frac{log q_{r}}{L} , 0, …, 0) + O(T) (3.62)

where the inner sum is over primes q_{j}. Since p_{1}, …, p_{r} are distinct primes, the primes q_{j} are then a permutation of p_{j}. Thus for some σ ∈ S_{r}, q_{j} = p_{σ(j)}, j = 1, …, r. Therefore

C_{r,r}(T) = \frac{1}{2π} κ(h)\frac{T}{(L)^{2r−1}} · ∑_{p_{1},...,p_{r}≤T} ∏_{j=1}^{r} \frac{|c(p_{j})|^{2}}{p_{j}}

∑_{σ∈S_{r}} Φ(−\frac{log p_{1}}{L} , …, −\frac{log p_{r}}{L} , \frac{log p_{σ(1)}}{L} , …, \frac{log p_{σ(r)}}{L} , 0, …, 0) + O(T) (3.63)

and we need no longer restrict to summing over distinct primes. The sum in (3.63) can be approximated by using summation by parts in the following form: by Proposition 2.3 (assuming H)

∑_{p≤X} \frac{|c(p)|^{2}}{p} = \frac{1}{2} log^{2} X + O(X)

Hence for reasonable functions ϕ (e.g. a C^{1} function)

∑_{p≤T} \frac{|c(p)|^{2}}{p} ϕ(\frac{log p}{L}) = ∫_{0}^{1/m} vϕ(v)dv · log^{2} T + O(log T) (3.64)

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 35

and therefore

C_{r,r}(T) = \frac{1}{2π} κ(h)\frac{T}{(L)^{2r−1}} · (L)^{2r} ∑_{σ_{∈}S_{r}} ∫_{0}^{1/m}⋯ ∫_{0}^{1/m} v_{1} … v_{r}

· Φ(−v_{1}, …, −v_{r}, v_{σ(1)}, …, v_{σ(r)}, 0, …, 0)dv_{1} … dv_{r} + O(T)

= κ(h)\frac{T}{2π} L ∑_{σ_{∈}S_{r}} ∫_{0}^{1/m}⋯ ∫_{0}^{1/m} v_{1} … v_{r}

· Φ(−v_{1}, …, −v_{r}, v_{σ(1)}, …, v_{σ(r)}, 0, …, 0)dv_{1} … dv_{r} + O(T)

(3.65)

as required.

If we use Lemma 3.10 and bring into account all possible choices of r = 0, …, [n/2] and signs, we obtain Theorem 3.1. Remark: The support condition |ξ_{1}| + ⋯ + |ξ_{n}| < 2/m. Consider first m = 1 and the case of ζ(s). In this case (and only in this case) there is an added term in the Explicit Formula (2.16) arising from the pole at s = 1, of the form

∏_{j=1}^{n} \{h(−\frac{i}{2T})T^{−mξ_{j}/2} + h(\frac{i}{2T})T^{mξ_{j}/2}\} ∼ h(0)^{n}T^{m(|ξ_{1}|+···+|ξ_{n}|)/2} (3.66)

Corresponding to this all the coefficients Λ(n) in the sum over primes are non-negative. From (3.66) it is clear that the region ∑_{j} |ξ_{j}| < 2, which appears in Theorem 1.2 is precisely that which renders (3.66) smaller than the main term T log T. Indeed, outside this region this polar term will be significant and the central diagonal terms in the proof of Theorem 1.2 are no longer dominant.

In the case of Dirichlet L-functions L(s, χ), the same is true but no longer because of the (non-existent) polar term - see below. For m ≥ 2 the region in which we prove Theorem 1.2 is no longer “natural”: In the case of the pair correlation function, the relevant sum coming from our analysis is

∑_{n_{1},n_{2}} \frac{c(n_{1})}{\sqrt{n_{1}}} \frac{c(n_{2})}{\sqrt{n_{2}}}g(T (2ξ_{1} log T + log n_{1}))g(T (2ξ_{2} log T − log n_{2}))f̂(ξ_{1}, ξ_{2})

(3.67)

with ξ_{1} + ξ_{2} = 0. If we assume that f̂ is supported near ξ_{1} = −ξ_{2} = ξ > 0 then the diagonal contribution is of size log T. Consider the off-diagonal contribution: These are essentially sums over primes

OD : = \frac{log^{2} T}{T^{2ξ}} ∑_{\begin{matrix}| log m/n|≪1/T \\ m≠n∼T^{2ξ} \end{matrix}} c(m)c(n)

= \frac{log^{2} T}{T^{2ξ}} ∑_{m∼T^{2ξ}} ∑_{1≤h≤T^{2ξ−1}} c(m)c(m + h)

≪ \frac{log^{2} T}{T^{2ξ}} ∑_{1≤h≤T^{2ξ−1}} ∑_{m∼T^{2ξ}} c(m)c(m + h)

(3.68)

The inner sum in (3.68) will have substantial cancellation if a(m) are the Fourier coefficients of a cusp form on GL(2)/Q. We expect the size of the inner sum to be of order square root of the number of terms - that is O(T^{ξ}). With the square root saving we find that the off-diagonal contribution will be bounded by

OD ≪ \frac{log^{2} T}{T^{2ξ}} ∑_{1≤h≤T^{2ξ−1}} T^{ξ} = T^{ξ−1} log^{2} T .  (3.69)

Hence for ξ < 1 (i.e. |ξ_{1}| + |ξ_{2}| < 2) we see that the diagonal still dominates. That is to say that the region should be the same as in the case m = 1.

In this respect we observe an important difference of the above analysis when m = 1 and a(p) = χ(p) is a Dirichlet character, say of conductor Q. Then the sum

∑_{X≤p≤2X} Λ(p)χ(p)Λ(p + 2kQ)χ(p + 2kQ)  (3.70)

is of order X, if we assume the “twin-prime” type conjectures of Hardy and Littlewood. Thus the inner sum of (3.68) does not have the same cancellation as for m = 2. In fact, as with ζ(s), the region |ξ_{1}|+|ξ_{2}| < 2 valid for L(s, χ), is the largest in which the diagonal dominates. Proof of Theorem 3.2: We first show that in Theorem 3.1 we can take h_{j} = χ_{[−1,1]} to be the characteristic function of the interval [−1, 1]. In what follows we assume the Riemann Hypothesis for L(s, π). In Theorem 3.1 we established that for h_{j} as in (3.5), and f satisfying

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 37

(3.6),

\frac{1}{N(T)} ∑ h_{1}(\frac{γ_{1}}{T}) … h_{n}(\frac{γ_{n}}{T})f(\frac{L}{2π} γ_{1}, …, \frac{L}{2π} γ_{n}) → κ(h)μ(f) (3.71)

where N(T) = #{0 < γ_{j} ≤ T } ∼ T L/2π, and μ(f) is the measure given by (3.9). By taking linear combinations, we obtain

\frac{1}{N(T)} ∑ h(\frac{γ⃗}{T})f(\frac{L}{2π} γ⃗) → κ(h)μ(f)  (3.72)

where h(r⃗) is a finite linear combination of functions of the form h_{1}(r_{1}) … h_{n}(r_{n}) and

κ(h) = ∫_{−∞}^{∞} h(r, …, r)dr  (3.73)

To extend the validity of (3.72) further, we note that if H(r⃗) is piecewise continuous of rapid decrease then given ϵ > 0, there are finite linear combinations h_{1}, h_{2} as above so that h_{1} ≤ H ≤ h_{2} and ∫_{−∞}^{∞}(h_{2} − h_{1})(r, …, r)dr < ϵ. We use these to show that (3.72) is valid for such H: Indeed, given f as above, we can find an f_{+} > 0 with |f| < f_{+} and f_{+} admissible for (3.72) (at least assuming Φ in (3.6) is C^{2}). If we set

D(H, f; T) := \frac{1}{N(T)} ∑ H(\frac{γ⃗}{T})f(\frac{L}{2π} γ⃗) − κ(H)μ(f) (3.74)

then

|D(H, f; T)| ≤ |D(h_{1}, f; T)| + |D(H − h_{1}, f; T)|

≤ |D(h_{1}, f; T)| + \frac{1}{N(T)} ∑(h_{2} − h_{1})(\frac{γ⃗}{T})f_{+}(\frac{L}{2π} γ⃗) + κ(H − h_{1})|μ(f)|

≤ |D(h_{1}, f; T)| + |D(h_{2} − h_{1}, f_{+}; T)| + κ(h_{2} − h_{1})μ(f_{+}) + κ(H − h_{1})|μ(f)|

and since (3.72) is valid for the first two terms, we find

lim sup_{T} |D(H, f; T)| ≤ κ(h_{2} − h_{1})μ(f_{+}) + κ(H − h_{1})|μ(f)| < ϵ(μ(f_{+}) + |μ(f)|)

Since ϵ > 0 is arbitrary, it follows that D(H, f; T) → 0.

With this approximation argument, we can include many more admissible functions h, and in particular the characteristic function of the cube [−1, 1]^{n}. Hence Theorem 3.1 is valid with h_{j}(r) = χ_{[−1,1]}(r) or χ_{[a,b]}, for a < b.

38 ZEÉV RUDNICK AND PETER SARNAK

Next, we need to discuss the passage from the normalization of zeros \frac{L}{2π}γ appearing in the definition of the smooth sums C_{n}(f, h, T) in (3.3), and the normalization γ̃ = \frac{m}{2π}γ log |γ| initially used to define C_{n}(f, T) in (3.2). We explain it for the pair correlation (n = 2). Consider for ψ(x) = f(x + y, y),

\begin{aligned}C̃(ψ, T) & = ∑_{T ≤γ_{1},γ_{2}≤2T} ψ(γ_{1} \frac{m log γ_{1}}{2π} − γ_{2} \frac{m log γ_{2}}{2π}) \\ C(ψ, T) & = ∑_{T ≤γ_{1},γ_{2}≤2T} ψ(\frac{L}{2π} (γ_{1} − γ_{2})) \end{aligned}  (3.75)

Then we claim that

C(ψ, T) − C̃(ψ, T) = O(T)  (3.76)

This will show that the different normalizations lead to the same main term, and thus prove Theorem 3.2.

To see this, observe that

C̃(ψ, T) = ∑_{T ≤γ_{1},γ_{2}≤2T} ψ(\frac{L}{2π} (γ_{1} − γ_{2}) + γ_{1} \frac{m}{2π} log \frac{γ_{1}}{T} − γ_{2} \frac{m}{2π} log \frac{γ_{2}}{T})

Applying the mean value theorem, (3.76) equals

2 ∑_{γ_{1}<γ_{2}} ψ′(δ_{γ_{1},γ_{2},T})(γ_{2} \frac{m}{2π} log \frac{γ_{2}}{T} − γ_{1} \frac{m}{2π} log \frac{γ_{1}}{T})

where δ_{γ_{1},γ_{2},T} is in the interval with endpoints \frac{m}{2π}(γ_{1} log γ_{1} − γ_{2} log γ_{2}) and \frac{L}{2π}(γ_{1} − γ_{2}). Thus (since 0 ≤ log \frac{γ_{j}}{T} ≤ log 2)

|C̃(ψ, T) − C(ψ, T)| ≪ ∑_{T ≤γ_{1}<γ_{2}≤2T} |ψ′(δ_{γ_{1},γ_{2},T})| · (γ_{2} − γ_{1}) (3.77)

For this fixed ψ, let ψ_{1} ≥ |ψ′| be a rapidly decreasing function on R which is even and monotone on [0, ∞). Then by (3.77)

|C̃(ψ, T) − C(ψ, T)| ≪ \frac{1}{L} ∑_{T ≤γ_{1},γ_{2}≤2T} ψ_{1}(\frac{L}{2π} (γ_{1} − γ_{2}))|\frac{L}{2π} (γ_{1} − γ_{2})|

We can find a majorant ψ_{+} and h which are admissible in Theorem 3.1, satisfying ψ_{+}(x) ≥ |x|ψ_{1}(x) and h ≥ χ_{[T,2T]}. Then with these choices we have

|C(ψ, T) − C̃(ψ, T)| ≪ \frac{1}{L} ∑ h(\frac{γ_{1}}{T})h(\frac{γ_{2}}{T})ψ_{+}(\frac{L}{2π})

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 39

and we can bound the sum by O(T L) by Theorem 3.1. Thus we find

|C(ψ, T) − C̃(ψ, T)| ≪ \frac{1}{L}T L ≪ T

as required. This establishes Theorem 3.2.

40 ZEÉV RUDNICK AND PETER SARNAK

4. Combinatorial Sieving

In §3 we showed that the unrestricted sums C_{n}(f, T) have a limiting distribution

C_{n}(f, T) := ∑_{i_{1},...,i_{n}} f(γ̃_{i_{1}}, …, γ̃_{i_{n}}) ∼ N(T) · ∫_{R^{n}} f̂(u)C\underline{O}(u)du (4.1)

where the sum is over all indices (i_{1}, …, i_{n}). However, the n-level correlation function R_{n}(f, T) is the same sum but over distinct indices; it differs from C_{n}(f, T) by omitting all sums over diagonals i_{j} = i_{k}, which measure lower order correlations. In this section we recover R_{n}(f, T) by a combinatorial sieving.

We begin with some set-theoretic combinatorics. A set partition \underline{F} of \underline{N} = \{1, …, n\} is a decomposition of \underline{N} into disjoint subsets [F_{1}, …, F_{ν}]. The collection Π_{n} of all set partitions of \underline{N} forms a lattice with the partial ordering given by \underline{F} ⪯ \underline{G} if every subset G_{i} is a union of subsets of \underline{F}. The minimal element of Π_{n} is \underline{O} = [\{1\}, \{2\}, …, \{n\}] and the maximal element is \underline{N} = \{1, 2, …, n\}.

The Möbius function of a poset such as Π_{n} is the unique function μ(x, y) so that for any functions f, g : Π_{n} → R, satisfying

f(x) = ∑_{x⪯y} g(y)  (4.2)

we have:

g(x) = ∑_{x⪯y} μ(x, y)f(y)  (4.3)

In the case of Π_{n}, the Möbius function can be computed explicitly ([15], §25), in particular

μ(\underline{O}, \underline{F}) = ∏_{j=1}^{ν(\underline{F})}(−1)^{|F_{j}|−1}(|F_{j}| − 1)!  (4.4)

Given a set partition \underline{F} = [F_{1}, …, F_{ν}] ∈ Π_{n}, we have an embedding ι\underline{F} : R^{ν} → R^{n}, given by ι\underline{F} (x_{1}, …, x_{ν}) = (y_{1}, …, y_{n}), where y_{l} = x_{j} if l ∈ F_{j}. For instance, if \underline{F} = [\{1, 3\}, \{2, 4\}] ∈ Π_{4} then ι\underline{F} (x, y) = (x, y, x, y).

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 41

We can define the lower order correlations

R\underline{F} (f, T) := R_{ν(\underline{F})}(ι∗ ; \underline{F} f, T) = ∑_{i_{1},...,i_{ν} ; distinct} f(ι\underline{F} (γ̃_{i_{1}}, …, γ̃_{i_{ν}})) (4.5)

This is the ν-level correlation between n zeros (γ_{1}, …, γ_{n}) where we mandate that the indices i_{l} = i_{k} are equal if l, k ∈ F_{j}, and i_{l} ≠ i_{k} if k, l are in different subsets of \underline{F}. Thus if \underline{F} = \underline{N} then R\underline{N}(f, T) = ∑ f(γ̃, …, γ̃) counts the number of zeros up to height T, while if \underline{F} = \underline{O} then R\underline{O} is just the n-level correlation function (1.3) defined in the Introduction (after division by N and where B_{N} = B_{N(T)}, N(T) = #{j : |γ_{j}| ≤ T }). We similarly define unrestricted correlation functions by setting

C\underline{F} (f, T) := C_{ν}(ι∗ ; \underline{F} f, T) = ∑_{i_{1},...,i_{ν}} f(ι\underline{F} (γ̃_{i_{1}}, …, γ̃_{i_{ν}}))  (4.6)

so that C_{n}(f, T) = C\underline{O}(f, T).

Observe that we have an identity

C\underline{O}(f, T) = ∑_{\underline{F}} R\underline{F} (f, T)  (4.7)

or more generally, for any \underline{G} ∈ Π_{n}

C\underline{G}(f, T) = ∑_{\underline{G}⪯\underline{F}} R\underline{F} (f, T)  (4.8)

This is merely partitioning the unrestricted sum for C\underline{O} as a sum over the various possibilities for coincidences between the indices. Thus we can use Möbius inversion to express the n-level correlation function R\underline{O}(f, T) in terms of the unrestricted sums C\underline{F} (f, T):

R\underline{O}(f, T) = ∑_{\underline{F}} μ(\underline{O}, \underline{F})C\underline{F} (f, T)  (4.9)

This allows us to find the limiting behaviour of R_{n}(f, T) by using Theorem 3.2 which yields the limit of C\underline{F} (f, T). To describe the answer, we define δ-functions as follows: For a subset S ⊂ {1, …, n}, we put

δ_{S}(u) = δ(∑_{l∈S} u_{l}) ,  (4.10)

where δ(x) = Dirac delta mass at the origin, and for a set partition \underline{F} = [F_{1}, …, F_{ν}] we define

δ\underline{F} (u) = ∏_{j=1}^{ν(\underline{F})} δ_{F_{j}}(u) .  (4.11)

δ\underline{F} is a delta function supported on the linear subspace

U\underline{F} = \{u ∈ R^{n} : ∑_{l∈F_{j}} u_{l} = 0,  j = 1, …, ν(\underline{F})\} . (4.12)

Lemma 4.1. With the assumptions as in Theorem 3.2

C\underline{F} (f, T) = N(T) ∫ Φ(u)C\underline{F} (u)du + O(T)

where C\underline{F} (u) is given by

C\underline{F} (u) = δ\underline{F} (u) + ∑_{r=1}^{[ν(\underline{F})/2]} ∑ ∏_{t=1}^{r} δ_{F_{i(t)}∪F_{j(t)}}(u)| ∑_{l∈F_{i(t)}} u_{l}| · ∏_{k≠i(a),j(b)} δ_{F_{k}}(u) (4.13)

the sum being over all possible choices of r pairs of subsets (F_{i(t)}, F_{j(t)}) of \underline{F}, no repetitions allowed.

Proof. We first express ι∗ ; \underline{F} f = Φ̂\underline{F} as a Fourier transform, in terms of the original Φ such that f = Φ̂:

Φ\underline{F} (v_{1}, …, v_{ν}) = δ(v_{1} + ⋯ + v_{ν}) ∫_{R^{n}} ∏_{j=1}^{ν} δ(v_{j} − ∑_{l∈F_{j}} u_{l})Φ(u)du (4.14)

Indeed,

ι∗ ; \underline{F} f(x_{1}, …, x_{ν}) = ∫_{R^{n}} Φ(ξ)δ(ξ_{1} + ⋯ + ξ_{n})e(−ξ · ι\underline{F} (x))dξ

= ∫_{ξ_{1}+···+ξ_{n}=0} Φ(ξ)e(− ∑_{j=1}^{ν} x_{j}( ∑_{l∈F_{j}} ξ_{l}))dξ

= ∫_{v_{1}+···+v_{ν}=0} {∫ ⋯ ∫ _{∑_{l∈F_{j}} ξ_{l}=v_{j}} Φ(ξ)dξ} e(− ∑_{j=1}^{ν} x_{j}v_{j})dv

which proves (4.14). Note that (4.14) implies that if Supp Φ ⊂ {∑_{j=1}^{n} |ξ_{j}| < r\} then Supp Φ\underline{F} ⊂ \{∑_{j=1}^{ν} |η_{j}| < r\}.

Now we use Theorem 3.2 to get the asymptotics of C\underline{F} (f, T) (note that the conditions TF 2,3 descend to ι∗ ; \underline{F} f):

C\underline{F} (f, T) = C_{ν}(ι∗ ; \underline{F} f, T) = N(T) · ∫_{R^{ν}} Φ\underline{F} (v)C_{ν}(v)dv + O(T)

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 43

On using (4.14) we find

∫_{R^{ν}} Φ\underline{F} (v)C_{ν}(v)dv = ∫_{R^{ν}} ∫_{R^{n}} Φ(u) ∏_{j=1}^{ν} δ(v_{j} − ∑_{l∈F_{j}} u_{l})C_{ν}(v_{1}, …, v_{ν})dudv

= ∫_{R^{n}} Φ(u)C_{ν}( ∑_{l∈F_{1}} u_{l}, …, ∑_{l∈F_{ν}} u_{l})du

Now substituting the expression (3.9) for C_{ν}(u) in Theorem 3.2 yields (4.13).

We have seen that

R\underline{O}(f, T) = N(T) ∫_{R^{n}} Φ(u)R\underline{O}(u)du + O(T)  (4.15)

where

R\underline{O}(u) = ∑_{\underline{F}} μ(\underline{O}, \underline{F})C\underline{F} (u)  (4.16)

In view of (4.15), (4.16) and the definition of the GUE determinant W_{n}(u) (1.5), Theorem 1.2 follows from

Theorem 4.1. Let W_{n}(x) = det(K(x_{i} − x_{j})), K(x) = \frac{sin πx}{πx}. Then for ∑_{j} |u_{j}| < 2, the Fourier transform Ŵ_{n}(u) is equal to R\underline{O}(u) in (4.16).

Similarly Theorem 1.1 follows from Theorem 3.1 (with h_{1} = ⋯ = h_{n} = h) and the above combinatorics.

We will divide the proof of Theorem 4.1 into three Propositions:

Proposition 4.1.

∑_{\underline{F}} μ(\underline{O}, \underline{F})C\underline{F} (u) = ∑_{\underline{F}} δ\underline{F} (u) ∏_{j=1}^{ν(\underline{F})} X_{F_{j}}(u)  (4.17)

where for any subset S ⊂ \underline{N},

X_{S}(u) = (−1)^{|S|−1}(|S| − 1)! + (−1)^{|S|−2} ∑_{S=S^{+}∪S^{−}}(|S^{+}| − 1)!(|S^{−}| − 1)!| ∑_{l∈S^{+}} u_{l}| . (4.18)

Proposition 4.2. The Fourier transform of the GUE determinant

Ŵ_{n}(u_{1}, …, u_{n}) = ∫_{R^{n}} det(K(x_{i} − x_{j}))e(∑_{j=1}^{n} u_{j}x_{j})dx

is given by

Ŵ_{n}(u) = ∑_{\underline{F}} δ\underline{F} (u) ∏_{j=1}^{ν(\underline{F})}(−1)^{|F_{j}|−1}Y_{F_{j}}(u)  (4.19)

where for any subset S ⊂ \underline{N}

Y_{S}(u) = ∑_{(i_{1},...,i_{m})} ∫_{−∞}^{∞} f_{2}(v)f_{2}(v + u_{i_{1}}) · · · · · f_{2}(v + u_{i_{1}} + ⋯ + u_{i_{m}})dv (4.20)

the sum over all cyclic permutations of S, and

f_{2}(v) = \{ \begin{matrix}1, & |v| ≤ 1/2 \\ 0, & |v| > 1/2 \end{matrix}  (4.21)

(so that K(x) = f̂_{2}(x) is the Fourier transform of f_{2}).

Proposition 4.3. For a subset S ⊂ \underline{N} with ∑_{l∈S} u_{l} = 0, ∑_{l∈S} |u_{l}| < 2, we have an identity

X_{S}(u) = (−1)^{|S|−1}Y_{S}(u)  (4.22)

By comparing the coefficients of δ\underline{F} (u) in Propositions 4.1 and 4.2, we see that Proposition 4.3 is exactly what is needed to establish Theorem 4.1. The rest of Section 4 is concerned with proving these Propositions. Proof of Proposition 4.1 It is expedient to introduce some structure on set partitions that corresponds to the combinatorics of pairs appearing in the calculations. To this end, we define a marking ϕ of a set partition \underline{G} = [G_{1}, …, G_{ν}] to be a choice of r ≥ 0 pairs of subsets (G_{i_{1}}^{+}, G_{i_{1}}^{−}), …, (G_{i_{r}}^{+}, G_{i_{r}}^{−}).

We will denote such a marking by

(\underline{G}, ϕ) = [G_{i_{1}}^{+}, G_{i_{1}}^{−}; G_{i_{2}}^{+}, G_{i_{2}}^{−}; …; G_{i_{r}}^{+}, G_{i_{r}}^{−}; G_{i_{2r+1}}^{o}; …, G_{i_{ν}}^{o} ] We will allow the trivial marking (\underline{G}, ϕ) = [G_{1}^{o}; …, G_{ν}^{o}].

A marking (\underline{G}, ϕ) reduces to an unmarked set partition \underline{F} if for all j = 1, …, r there is a subset F_{j} of \underline{F} so that F_{j} = G_{i_{j}}^{+} ∪ G_{i_{j}}^{−}, and likewise for all k there is a subset of \underline{F} so that G_{k}^{o} = F_{i(k)}. We will denote this symbolically as \underline{F} = red(\underline{G}, ϕ) or (\underline{G}, ϕ) → \underline{F}. Note: A marked set partition reduces to a unique unmarked one; however there are usually several different marked partitions reducing to the same unmarked one.

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 45

With the notion of marking of a set partition, we can describe the formula (4.13) for C\underline{G}(u) as

C\underline{G}(u) = ∑_{ϕ} δ_{red(\underline{G},ϕ)}(u) ∏_{j=1}^{r} | ∑_{l∈G_{j}^{+}} u_{l}|  (4.23)

where the sum is over all possible marking of \underline{G}, and for the trivial marking the empty product is interpreted as equal to 1. Therefore we find

\begin{aligned}∑_{\underline{G}} μ(\underline{O}, \underline{G})C\underline{G}(u) & = ∑_{(\underline{G},ϕ)} μ(\underline{O}, \underline{G})δ_{red(\underline{G},ϕ)}(u) ∏_{j=1}^{r} | ∑_{l∈G_{j}^{+}} u_{l}| \\ & = ∑_{\underline{F}} δ\underline{F} (u) ∑_{(G,ϕ)→\underline{F}} μ(\underline{O}, \underline{G}) ∏_{j=1}^{r} | ∑_{l∈G_{j}^{+}} u_{l}| \end{aligned} (4.24)

Now fix a set partition \underline{F}, and consider the coefficient of δ\underline{F} above. Then we have a factorization identity: If u ∈ U\underline{F} then

\begin{matrix}∑_{(G,ϕ)→\underline{F}} μ(\underline{O}, \underline{G}) ∏_{j=1}^{r} | ∑_{l∈G_{j}^{+}} u_{l}| \\ = ∏_{j=1}^{ν(\underline{F})} \{(−1)^{|F_{j}|−1}(|F_{j}| − 1)! + (−1)^{|F_{j}|−2} ∑_{F_{j}=G_{j}^{+}∪G_{j}^{−}}(|G_{j}^{+}| − 1)!(|G_{j}^{−}| − 1)!| ∑_{l∈G_{j}^{+}} u_{l}|\} . \end{matrix} (4.25)

To prove this, just multiply out the RHS of (4.25) and compare with (4.4). The factors in the product are exactly what we called X_{F_{j}} in Proposition 4.1, and so we find as desired that

∑_{\underline{G}} μ(\underline{O}, \underline{G})C\underline{G}(u) = ∑_{\underline{F}} δ\underline{F} (u) ∏_{j=1}^{ν(\underline{F})} X_{F_{j}}(u)  (4.26)

This proves Proposition 4.1. Proof of Proposition 4.2: We expand the determinant W_{n} as a sum over all permutations of \underline{N} = \{1, …, n\}:

W_{n}(x_{1}, …, x_{n}) = ∑_{σ∈S_{n}}(−1)^{σ} ∏_{j=1}^{n} K(x_{j} − x_{σ(j)})  (4.27)

When taking the Fourier transform of W_{n} decompose each permutation as a product over disjoint cycles σ = τ_{1} · · · · · τ_{r}, with τ_{j} = (i_{1}, …, i_{m}) a cycle of length m = m(j). Note that (−1)^{σ} = ∏(−1)^{m(j)−1}. We then

46 ZEÉV RUDNICK AND PETER SARNAK

notice that the Fourier transform of each summand in (4.27) breaks up into a product of Fourier transforms of similar expressions over the cycles τ_{j}:

∫ ∏_{j=1}^{n} K(x_{j} − x_{σ(j)})e(∑ u_{j}x_{j})dx

= ∏_{j=1}^{r} ∫ K(x_{i_{1}} − x_{i_{2}})K(x_{i_{2}} − x_{i_{3}}) … K(x_{i_{m}} − x_{i_{1}})e(u_{i_{1}}x_{i_{1}} + ⋯ + u_{i_{m}}x_{i_{m}})dx .

It is therefore sufficient to compute the Fourier transform of each factor separately when we have a cyclic permutation.

Lemma 4.2. ∫ K(x_{1} − x_{2})K(x_{2} − x_{3}) … K(x_{m} − x_{1})e(u_{1}x_{1} + ⋯ + u_{m}x_{m})dx =

δ(u_{1} + ⋯ + u_{m}) ∫_{−∞}^{∞} f_{2}(v)f_{2}(v + u_{1}) … f_{2}(v + u_{1} + ⋯ + u_{m−1})dv

Proof. Set t_{j} = x_{j} − x_{j+1}, j = 1, …, m − 1. Then changing variables we find

∫ K(x_{1} − x_{2})K(x_{2} − x_{3}) … K(x_{m} − x_{1})e(u_{1}x_{1} + ⋯ + u_{m}x_{m})dx
= ∫ K(t_{1}) … K(t_{m−1})K(−t_{1} ⋯ − t_{m−1})
e (t_{1}u_{1} + ⋯ + t_{m−1}(u_{1} + ⋯ + u_{m−1}) + x_{m}(u_{1} + ⋯ + u_{m})) dt_{1} … dt_{m−1}dx_{m}
= δ(u_{1} + ⋯ + u_{m}) ∫ K(t_{1}) … K(t_{m−1})K(−t_{1} ⋯ − t_{m−1})e(t_{1}s_{1} + ⋯ + t_{m−1}s_{m−1})dt

where we have set s_{k} = u_{1} + ⋯ + u_{k}.

Since K = f̂_{2}, we can use Parseval to see ∫_{−∞}^{∞} K(t_{1})K(−τ − t_{1})e(t_{1}s_{1})dt_{1} = ∫_{−∞}^{∞} f_{2}(v)f_{2}(v + s_{1})e(vτ)dv (4.28)

Applying this with τ = t_{2} + ⋯ + t_{m−1}, we find:

∫ K(t_{1}) … K(t_{m−1})K(−t_{1} ⋯ − t_{m−1})e(t_{1}s_{1} + ⋯ + t_{m−1}s_{m−1})dt =

∫ f_{2}(v)f_{2}(v + s_{1})e(v(t_{2} + ⋯ + t_{m−1}))K(t_{2}) … K(t_{m−1})e(t_{2}s_{2} + ⋯ + t_{m−1}s_{m−1})dtdv =

∫ f_{2}(v)f_{2}(v + s_{1}) ∏_{j=2}^{m−1} ∫ K(t_{j})e(t_{j}(v + s_{j}))dt_{j}dv =

∫ f_{2}(v)f_{2}(v + s_{1})f_{2}(v + s_{2}) … f_{2}(v + s_{m−1})dv .

To finish the proof of Proposition 4.2, we decompose the permutations into products of disjoint cyclic permutations indexed by set partitions \underline{F} of \underline{N} = \{1, …, n\}:

S_{n} = ∐_{\underline{F}} S^{∗}(F_{1}) × ⋯ × S^{∗}(F_{ν})

where for a subset F_{j} ⊂ \underline{N}, S^{∗}(F_{j}) denotes the set of all cyclic permutations of the indices in F_{j}. The sign of any cyclic permutation in S^{∗}(F_{j}) is (−1)^{|F_{j}|−1}. Proposition 4.2 now follows. Proof of Proposition 4.3: It clearly suffices to prove Proposition 4.3 for the case S = \underline{N}, which we assume from now on. We will need some preparation: For u ∈ R^{n}, with ∑_{j} u_{j} = 0, and ∑_{j} |u_{j}| < 2, and an ordering θ = (θ(1), …, θ(n)) of \underline{N}, we define the consecutive partial sums

s_{k}(θ) = ∑_{j=1}^{k} u_{θ(j)} = u_{θ(1)} + ⋯ + u_{θ(k)}  (4.29)

and let M(θ), m(θ) be the maximum (respectively, the minimum) of these partial sums:

\begin{aligned}M_{u}(θ) & = max\{s_{k}(θ), k = 1, …, n\} , \\ m_{u}(θ) & = min\{s_{k}(θ), k = 1, …, n\} . \end{aligned}  (4.30)

Further set

V (θ) = M(θ) − m(θ)  (4.31)

When no ordering θ is explicitly given, we just write V for the corresponding quantity.

It is useful to think of u_{θ(j)} as the increments of a “random walk” on the real line, starting and ending at the origin, and the partial sums s_{k}(θ) are the positions after k steps. M and m are the farthest positions to the right (respectively, to the left), and the difference V (θ) = M(θ)− m(θ) is then the maximal deviation of the walk. From this description it is apparent that V (θ) = V (θ′) if θ′ is a rotation of θ, e.g. θ′ = (θ(2), …, θ(n), θ(1)).

The connection of this to our previous discussion is

Lemma 4.3. If u_{1} + ⋯ + u_{n} = 0, ∑_{j} |u_{j}| < 2 then

∫_{−∞}^{∞} f_{2}(v)f_{2}(v + u_{1}) … f_{2}(v + u_{1} + ⋯ + u_{n})dv = 1 − V . (4.32)

Proof. Recall that f_{2}(v) is the characteristic function of the interval I = [−1/2, 1/2]. Thus the integral (4.32) is the length of the intersection of the intervals I, −s_{1} + I, ⋯ − s_{n−1} + I. This intersection is non-empty if ∑ |u_{j}| < 2, in which case it equals the interval [−m−1/2, −M +1/2] whose length is −M + 1/2 − (−m − 1/2) = 1 − V.

Set

T (u) = ∑_{{θ}}^{′} V (θ)  (4.33)

the sum being taken over all orderings modulo rotations - i.e. over cyclic permutations of {1, …, n}, of which there are (n − 1)!. Using Lemma 4.3, we can rewrite Y\underline{N}(u) in (4.20) as

Y\underline{N}(u) = (n − 1)! − T (u)  (4.34)

Taking into account the definition of X_{S}(u) (4.18), in order to prove Proposition 4.3, it suffices to prove the following identity between piecewise linear functions

T (u) = ∑_{[F,F^{c}]}(|F | − 1)!(n − |F | − 1)!| ∑_{l∈F} u_{l}|  (4.35)

This we accomplish below by adapting Spitzer’s combinatorial method [34] ^{3}.

Both sides of (4.35) are continuous in u, so it suffices to prove (4.35) for u generic, i.e. the components u_{i} are linearly independent over the rationals. Since V (θ) is invariant under rotations, we may write T (u) in 4.33 as

T (u) = \frac{1}{n} ∑_{θ} V (θ) = \frac{1}{n} ∑_{θ} M(θ) − \frac{1}{n} ∑_{θ} m(θ)  (4.36)

3Apparently this identity is quite old - it can already be found in Kac [12]

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 49

The sums ∑_{θ} M(θ) and ∑_{θ} m(θ) are easily seen to be negatives of each other, and so

T (u) = \frac{2}{n} ∑_{θ} M(θ)  (4.37)

For u = (u_{1}, …, u_{r}) ∈ R^{r}, define the (polygonal) walk W_{u} to be the walk (0, 0) → (1, u_{1}) → ⋯ → (r, u_{1} + ⋯ + u_{r}). The chord is the segment connecting (0, 0) to (r, u_{1} + ⋯ + u_{r}). Define the upper convex envelope U_{u} of the walk W_{u} to be the lowest convex curve lying above W_{u} (Figure 2).

Lemma 4.4 (Spitzer [34], Thm 2.1). Given u = (u_{1}, …, u_{r}) in generic position, there is a unique rotation of u so that the walk lies below its chord.

Given u, W_{u} and U_{u} let 0 < k_{1} < k_{2} < ⋯ < k_{ν} = r be the first coordinates of the vertices of the upper convex walk U_{u}. Note that the walk W_{u} restricted to the sets {1, …, k_{1}}, {k_{1} + 1, …, k_{2}}, … have the properties

SP 1. The walks lie below their chords.

SP 2. Their respective slopes are decreasing, i.e.

\frac{u_{1} + ⋯ + u_{k_{1}}}{k_{1}} > \frac{u_{k_{1}+1} + ⋯ + u_{k_{2}}}{k_{2} − k_{1}} > …

Returning to the ordering θ of \underline{N} = \{1, …, n\}, and u ∈ R^{n}, ∑_{j} u_{j} = 0, which we assume is generic, we get a convex polygon U_{u}(θ) above the walk W_{u}(θ). This convex polygon starts at (0, 0) and ends at (n, 0) (Figure 2). It determines intervals as above G_{1}(θ) = [1, …, k_{1}], G_{2}(θ) = [k_{1}+1, …, k_{2}], …. Each of these come with an ordering ψ_{j}(θ). So for each ordering θ we get data D(θ) = (G_{1}(θ), ψ_{1}(θ)), (G_{2}(θ), ψ_{2}(θ)), … satisfying SP 1 and SP 2. Moreover

M_{u}(θ) = \frac{1}{2} ∑_{j} | ∑_{l∈G_{j}(θ)} u_{l}|  (4.38)

Conversely, given u generic and a set partition \underline{G} = [G_{1}, …, G_{ν}] of \underline{N} and orderings ψ_{j} of G_{j}, we can uniquely arrange them to satisfy SP 1

50 ZEÉV RUDNICK AND PETER SARNAK

and SP 2 by first arranging their slopes (i.e. SP 2) and then using Spitzer’s Lemma 4.4 to adjust each ordering ψ_{j} by a (unique) rotation so as to satisfy SP 1. That is, using this bijection and (4.38) we have

\begin{aligned}∑_{θ} M_{u}(θ) & = \frac{1}{2} ∑_{\underline{G}=[G_{1},...,G_{ν}]}(|G_{1}| − 1)! · · · · · (|G_{ν}| − 1)! (| ∑_{l∈G_{1}} u_{l}| + ⋯ + | ∑_{l∈G_{ν}} u_{l}|) \\ & = \frac{1}{2} ∑_{F_{1}⊂\underline{N}}(|F_{1}| − 1)!| ∑_{l∈F_{1}} u_{l}| · (  ∑_{\underline{F} =[F_{1},F_{2},...,F_{ν}]}(|F_{2}| − 1)! … (|F_{ν}| − 1)!) \end{aligned} (4.39)

For fixed subset F_{1} ⊂ \underline{N}, the innermost sum is clearly equal to the sum over all set partitions of the complement F_{1}^{c} = \underline{N} − F_{1} of

∑_{[F_{2},...,F_{ν}]}(|F_{2}| − 1)! … (|F_{ν}| − 1)!  (4.40)

This counts all the permutations of F_{1}^{c} when writing a permutation as a product of disjoint cycles. So the sum (4.40) is simply |F_{1}^{c}|! = (n − |F_{1}|)!. Hence

∑_{θ} M_{u}(θ) = \frac{1}{2} ∑_{F_{1}⊂\underline{N}}(|F_{1}| − 1)!(n − |F_{1}|)!| ∑_{l∈F_{1}} u_{l}|  (4.41)

Grouping together the terms corresponding to F_{1} and its complement F_{1}^{c} and using ∑_{j} u_{j} = 0 (so that | ∑_{l∈F_{1}} u_{l}| = | ∑_{l∈F_{1}^{c}} u_{l}|) we have

\begin{aligned}∑_{θ} M_{u}(θ) & = \frac{1}{2} ∑_{[F,F^{c}]} ((|F | − 1)!(n − |F |)! + (n − |F | − 1)!|F |!) | ∑_{l∈F} u_{l}| \\ & = \frac{n}{2} ∑_{[F,F^{c}]}(|F | − 1)!(n − |F | − 1)!| ∑_{l∈F} u_{l}| \end{aligned} (4.42)

Hence from (4.37)

T (u) = ∑_{[F,F^{c}]}(|F | − 1)!(n − |F | − 1)!| ∑_{l∈F} u_{l}|  (4.43)

This concludes the proof of (4.35) and so of Proposition 4.3.

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 51

5. Appendix

5.1. Our goal in this appendix is two-fold. Firstly, we describe the local L-factors at ramified places and secondly we derive the estimates (2.3) and (2.5) for the α_{π}(j, p) and μ_{π}(j).

Let π_{p} to be an irreducible admissible (generic) representation of G = GL_{m}, over R or Q_{p}, with unitary central character. From the description below it follows that if π_{p} is tempered then L(s, π_{p}) is holomorphic in Re s > 0. The general non-tempered representation π_{p} can be described as a Langlands quotient: There is a standard parabolic subgroup P of type (m_{1}, …, m_{r}) (so that the Levi complement M_{P} = P/U_{P} ≃ GL(m_{1}) × ⋯ × GL(m_{r})), tempered representations σ_{j} of GL(m_{j}) and real numbers t_{1} > ⋯ > t_{r} (called the Langlands parameters of π_{p}) so that

π_{p} = J(G, P; σ_{1}[t_{1}], … σ_{r}[t_{r}])  (5.1)

is the unique irreducible quotient of the induced representation Ind(G, P; σ_{1}[t_{1}], … σ_{r}[t_{r}]). Here the twist operation for t ∈ C is defined as σ[t] := σ ⊗ | det(·)|^{t}. If π_{p} is unitary, then {σ_{j}[t_{j}]} = {σ̃_{k}[−t_{k}]}. In terms of this data, the L-factor is given by

L(s, π_{p}) = ∏_{j=1}^{r} L(s + t_{j}, σ_{j})  (5.2)

where the definition of the tempered factors still needs to be given. Indeed in this case, for σ tempered, there is a standard parabolic P of type (m_{1}, …, m_{r}) and square-integrable representations τ_{j} of GL(m_{j}) so that σ is isomorphic to the full induced representation I(G, P; τ_{1}, …, τ_{r}). Then

L(s, σ) = ∏_{j=1}^{r} L(s, τ_{j})  (5.3)

This reduces the problem to the case of square-integrable representations. We describe these separately in the p-adic and real setting.

5.2. p-adic factors. For the p-adic case, the square-integrable representations are built out of super-cuspidal representations as follows: If

52 ZEÉV RUDNICK AND PETER SARNAK

π is a unitary square-integrable of GL_{m}(Q_{p}), then there is a divisor d|m, a standard parabolic subgroup P of type (d, …, d) and a unitary supercuspidal representation ρ of GL_{d} so that π is the unique square-integrable constituent of Ind(G, P; ρ_{1}, …, ρ_{n}), where n = m/d, and ρ_{j} = ρ ⊗ | det |^{j−(n+1)/2}, j = 1, …, n. We will write π = Δ(n, ρ). We note that the contragredient of such a representation is Δ̃(n, ρ) = Δ(n, ρ̃). Then the principal L-factor is given by L(s, π) = L(s + (n − 1)/2, ρ). Next, for supercuspidal representations ρ, we have L(s, ρ) = 1 unless m = 1 (in which case the supercuspidal condition is empty) and ρ = |·|^{t} is unramified, in which case L(s, ρ) = (1−p^{−(s+t)})^{−1}. This completes the description of the p-adic factors for the principal L-function.

Next we describe the p-adic factors for the Rankin-Selberg L-function [8]. Let π, π′ be irreducible, unitary generic representations of GL_{m}(Q_{p}) and GL_{m′}(Q_{p}), respectively. We say two representations π_{1}, π_{2} are in the same twist class if deg π_{1} = deg π_{2} and π_{1} = π_{2} ⊗ | det |^{t}. First, if π is supercuspidal then for deg π ≥ deg π′, the Rankin-Selberg local factor L(s, π × π′) equals 1 unless π′ is twist equivalent to π̃: π′ = π̃[t] (in particular π′ is itself supercuspidal). In that case

L(s, π × π̃[t]) = L(s + t, π × π̃) = \frac{1}{1 − p^{−r(t+s)}}  (5.4)

where r| deg π is the order of the cyclic group of unramified characters χ = | det |^{u} for which π ⊗ χ ≃ π.

Now if π = Δ(n, ρ), π′ = Δ(n′, ρ′) are square-integrable representations of GL_{m} and GL_{m′} respectively, then

L(s, Δ(n, ρ) × Δ(n′, ρ′)) = ∏_{j=1}^{min(n,n′)} L(s + \frac{n + n′}{2} − j, ρ × ρ′) (5.5)

In particular we see that L(s, π × π̃′) = 1 unless ρ, ρ′ are in the same twist class.

For the general π_{p} given as a Langlands quotient π = J(G, P; σ_{1}[t_{1}], …, σ_{r}[t_{r}]) as in (5.1), we can by using induction by stages assume that σ_{j} =

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 53

Δ(n_{j}, ρ_{j}) and t_{1} ≥ ⋯ ≥ t_{r}. Then we have

\begin{aligned}L(s, π × π̃) & = ∏_{j,k=1}^{r} L(s + t_{j} − t_{k}, σ_{j} × σ̃_{k}) \\ & = ∏_{j,k=1}^{r} ∏_{ν=1}^{min(n_{j},n_{k})} L(s + t_{j} − t_{k} + \frac{n_{j} + n_{k}}{2} − ν, ρ_{j} × ρ̃_{k}) \end{aligned} (5.6)

We know that the only factors which contribute are those for which ρ_{j}, ρ_{k} are in the same twist class. We can partition the components ρ_{1}, …, ρ_{r} into twist classes; there there is a set partition \underline{F} = [F_{1}, …, F_{h}] of {1, …, r} and unitary supercuspidals ρ^{l}, l = 1, …, h, which are twist-inequivalent, and reals u_{j} so that for j ∈ F_{l}, ρ_{j} = ρ^{l}[iu_{j}] is a unitary twist of ρ^{l}. We then find

L(s, π × π̃) = ∏_{l} ∏_{j,k∈F_{l}} L(s + t_{j} − t_{k}, Δ(n_{j}, ρ^{l}[iu_{j}]) × Δ̃(n_{k}, ρ^{l}[iu_{k}])) =: ∏_{l} L_{F_{l}}(s) (5.7)

We have

\begin{aligned}L_{F_{l}}(s) & = ∏_{j,k∈F_{l}} L(s + t_{j} − t_{k}, Δ(n_{j}, ρ^{l}[iu_{j}]) × Δ̃(n_{k}, ρ^{l}[iu_{k}])) \\ & = ∏_{j,k∈F_{l}} ∏_{ν=1}^{min n_{j},n_{k}} L(s + s_{j} − s_{k} + \frac{n_{j} + n_{k}}{2} − ν, ρ^{l} × ρ̃^{l}) \end{aligned} (5.8)

where s_{j} = t_{j} + iu_{j}. We know that L(s, ρ^{l} × ρ̃^{l}) = (1 − p^{−r_{l}s})^{−1} for suitable r_{l}| deg ρ^{l} (5.4), and so we find

L_{F_{l}}(s) = ∏_{j,k∈F_{l}} ∏_{ν=1}^{min(n_{j},n_{k})}(1 − (p^{ν−(n_{j}+n_{k})/2−s_{j}+s_{k}}p^{−s})^{r_{l}})^{−1} (5.9)

To transform this further, we note that by unitarity, the set {s_{j} : j ∈ F_{l}} is stable under s ↦ −s̄. Thus

\begin{aligned}L_{F_{l}}(s) & = ∏_{j,k∈F_{l}} ∏_{ν=1}^{min(n_{j},n_{k})}(1 − (p^{ν−(n_{j}+n_{k})/2−s_{j}−s̄_{k}}p^{−s})^{r_{l}})^{−1} \\ & = ∏_{j,k∈F_{l}} ∏_{ν=1}^{min(n_{j},n_{k})}(1 − (p^{ν}z_{j}z̄_{k})^{r_{l}}p^{−r_{l}s})^{−1} \end{aligned} (5.10)

where

z_{j} = p^{−(n_{j}/2+s_{j})},  j ∈ F_{l}  (5.11)

To conclude, we see that with the above notation,

L(s, π × π̃) = ∏_{l} ∏_{j,k∈F_{l}} ∏_{ν=1}^{min(n_{j},n_{k})}(1 − (p^{ν}z_{j}z̄_{k}p^{−s})^{r_{l}})^{−1} (5.12)

Therefore if t_{1} > ⋯ > t_{r} are the Langlands parameters of π (5.1), then the local factor L(s, π_{p} ×π̃_{p}) is holomorphic for Re s > 2 max{|t_{j}|} with a pole at s = 2 max{|t_{j}|}. Since it is known that the local factor is holomorphic in Re s ≥ 1 [8], being the g.c.d. of local integrals with this property, we see that for π_{p} generic and unitary the Langlands parameters satisfy max{|t_{j}|} < 1/2. As a consequence, if we write the principal local factor as L(s, π_{p}) = ∏_{j=1}^{m}(1 − α(p, j)p^{−s})^{−1}, then we see that |α(p, j)| ≤ p^{max{|t_{j}|}} < p^{1/2}. This proves (2.2) for all p < ∞.

We can expand L(s, π_{p} × π̃_{p}) and L_{F_{l}}(s) in a Dirichlet series

\begin{aligned}L(s, π_{p} × π̃_{p}) & = ∑_{e=0}^{∞} \frac{b(p^{e})}{p^{es}} \\ L_{F_{l}}(s) & = ∑_{e=0}^{∞} \frac{b_{l}(p^{er_{l}})}{p^{er_{l}s}} \end{aligned}  (5.13)

Lemma 5.1. If r_{l}|e then

b_{l}(p^{e}) ≥ \frac{1}{e}| ∑_{j∈F_{l}}(p^{(1−n_{j})/2−s_{j}})^{e}|^{2}  (5.14)

Proof. Take logarithms in (5.10) to find

log L_{F_{l}}(s) = ∑ \frac{p^{−er_{l}s}}{e} λ_{l}(e)

with

\begin{aligned}λ_{l}(e) & = ∑_{j,k∈F_{l}} ∑_{ν=1}^{min(n_{j},n_{k})}(p^{ν}z_{j}z̄_{k})^{er_{l}} \\ & = ∑_{1≤ν≤max\{n_{j}:j∈F_{l}\}}(p^{ν})^{er_{l}}| ∑_{j∈F_{l}:n_{j}≥ν} z_{j}^{er_{l}}|^{2} \end{aligned}

On substituting z_{j} = p^{−(n_{j}/2−s_{j})}, we get

\begin{aligned}λ_{l}(e) & = ∑_{1≤ν≤max\{n_{j}:j∈F_{l}\}} | ∑_{j∈F_{l}:n_{j}≥ν}(p^{(ν−n_{j})/2−s_{j}})^{er_{l}}|^{2} \\ & ≥ | ∑_{j∈F_{l}}(p^{(1−n_{j})/2−s_{j}})^{er_{l}}|^{2} \end{aligned}

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 55

Exponentiating, we find that

b_{l}(p^{er_{l}}) ≥ \frac{λ_{l}(e)}{e} ≥ \frac{1}{e}| ∑_{j∈F_{l}}(p^{(1−n_{j})/2−s_{j}})^{er_{l}}|^{2}

As a consequence of (2.19) and (5.14) we see that the L(s, π × π̃) has non-negative coefficients b(n) ≥ 0.

5.3. Archimedean factors. Any irreducible unitary representation of G = GL_{m}(R) is given as a Langlands quotient

π_{∞} = J(G, P; σ_{1}[s_{1}], …, σ_{r}[s_{r}])  (5.15)

where σ_{j} are square-integrable representations and Re s_{1} ≥ ⋯ ≥ Re s_{j}. For the real case, GL_{n}(R) does not have square-integrable representations if n ≥ 3. To describe the principal L-factors in the remaining cases n = 1, 2, we define Gamma factors by

\begin{aligned}Γ_{R}(s) & = π^{−s/2}Γ(\frac{s}{2}) \\ Γ_{C}(s) & = 2(2π)^{−s}Γ(s) \end{aligned}  (5.16)

Note that the duplication formula reads Γ_{C}(s) = Γ_{R}(s)Γ_{R}(s + 1). In the case n = 1 (when the condition to be square-integrable is vacuous), the unitary representations are of the form π(x) = |x|^{t} or π(x) = sign(x)|x|^{t} = |x|^{t+1}x^{−1}, with t ∈ iR. We then set L(s, π_{∞}) = Γ_{R}(s + t) in the former case, and L(s, π_{∞}) = Γ_{R}(s + t + 1) in the latter. In case n = 2, the unitary square-integrable representations are unitary twists of the k-th discrete series D_{k}, k ≥ 2 (these correspond to holomorphic forms of weight k), for which the L-factor is given by L(s, D_{k}) = Γ_{C}(s+ \frac{k−1}{2}).

To summarize, for p = ∞ the local factor has the following form: We can write m = r_{1} + 2r_{2}, and there are complex numbers s_{j} = t_{j} + iu_{j}, j = 1, …, r_{1} +r_{2} satisfying ∑_{j=1}^{r_{1}} t_{j} +2 ∑_{j=1}^{r_{2}} t_{j} = 0 and integers k_{j} ≥ 2, j = 1, …, r_{2} so that

L(s, π_{∞}) = ∏_{j=1}^{r_{1}} Γ_{R}(s + s_{j}) ∏_{j=1}^{r_{2}} Γ_{C}(s + s_{r_{1}+j} + \frac{k_{j} − 1}{2}) (5.17)

56 ZEÉV RUDNICK AND PETER SARNAK

We can rewrite this using the duplication formula as a product of m real Gamma factors:

L(s, π_{∞}) = ∏_{j=1}^{m} Γ_{R}(s + μ_{π}(j))  (5.18)

The Rankin-Selberg local factors are defined in terms of L-factors attached to representations of the Weil group W_{R} by means of the Langlands correspondence, and L(s, π_{∞} × π̃_{∞}) can be computed from knowledge of a few basic cases. We will list them and use this knowledge to derive (2.5). Suppose that π_{∞} is a unitary irreducible generic representation of GL_{m}(R). If we exhibit π_{∞} as a Langlands quotient J(G, P; σ_{1}[s_{1}], …, σ_{r}[s_{r}]) as in (5.15), then

L(s, π_{∞} × π̃_{∞}) = ∏_{j,k} L(s + s_{j} − s_{k}, σ_{j} × σ̃_{k})  (5.19)

It therefore suffices to check the following cases:

1. L(s, σ × 1) = L(s, σ);

2. L(s, sign × sign) = L(s, 1) = Γ_{R}(s);

3. D_{k} ⊗ sign ≃ D_{k} and so L(s, D_{k} × sign, s) = L(s, D_{k}) = Γ_{C}(s + (k − 1)/2);

4. If k_{1} ≥ k_{2} then L(s, D_{k_{1}} × D_{k_{2}}) = Γ_{C}(s + (k_{1} − k_{2})/2)Γ_{C}(s + (k_{1} +

k_{2})/2 − 1). In particular, L(s, π_{∞} × π̃_{∞}) has its first pole at s = 2 max |t_{j}|, t_{j} = Re s_{j}. On the other hand if π_{∞} is the local component of a cuspidal automorphic representation π then according to RS 3 of Section 2, L(s, π_{∞}×π̃_{∞})L(s, π×π̃) is holomorphic for Re s > 1 with a simple pole at s = 1 coming from the Euler product L(s, π×π̃). Thus L(s, π_{∞}×π̃_{∞}) is holomorphic for real s ≥ 1 and hence 2 max |t_{j}| < 1. That is, in the notations of (5.18), Re μ_{π}(j) > −1/2, proving (2.5). Remark: We have given a global proof of (2.5) in order to avoid the complications involved in carrying out the analogue of the p-adic proof via the archimedean Rankin-Selberg integrals [11]. A purely local proof of (2.5) is given in [1] by going through Vogan’s classification of the unitary dual of GL_{m}(R) and checking which representations are

ZEROS OF PRINCIPAL L-FUNCTIONS AND RANDOM MATRIX THEORY 57

generic. On the other hand, the improvements of the bounds (2.2) and (2.5) described below, do require global considerations.

5.4. Global bounds. The first improvement is for the finite places and follows a well known argument [31].

Proposition 5.1. Let π be an irreducible cuspidal automorphic representation of GL_{m}/Q, and let π_{p} be a local constituent of π. Then

|α_{π}(p, j)| ≤ p^{1/2−1/(m^{2}+1)}  (5.20)

Proof. This is merely an application of Landau’s Theorem [13]. Indeed we have shown that the series L(s, π×π̃) = ∑ b(n)n^{−s} has non-negative coefficients, and by RS 3 a meromorphic continuation with Gamma factors and poles exactly at s = 0, 1. In Landau’s notation we have β = 1 and η = m^{2}/2 and his result gives

∑_{n≤x} b(n) = Ax + O_{ϵ}(x^{\frac{2η−1}{2η+1} +ϵ})

for all ϵ > 0. Moreover, since b(n) ≥ 0, it follows that

b(n) ≪_{ϵ} n^{\frac{2η−1}{2η+1} +ϵ}

For π_{p} unramified, in view of (2.19), this means that for e ≥ 1,

| ∑_{j=1}^{m} α_{π}(p, j)^{e}|^{2} = |a_{π}(p^{e})|^{2} ≤ eb(p^{e}) ≪_{ϵ} ep^{e( \frac{2η−1}{2η+1} +ϵ)}

This clearly implies that

|α_{π}(p, j)| ≤ p^{\frac{1}{2} ( \frac{2η−1}{2η+1} )} = p^{\frac{1}{2} − \frac{1}{m^{2}+1}}

For ramified primes we find from (5.14) that for all l, if r_{l}|e then

| ∑_{j∈F_{l}}(p^{(1−n_{j})/2−s_{j}})^{e}|^{2} ≤ eb(p^{e}) ≪_{ϵ} ep^{2e( \frac{1}{2} − \frac{1}{m^{2}+1} +ϵ)}

and therefore we get a bound on the Langlands parameters

max_{j} |t_{j}| ≤ \frac{1}{2} − \frac{1}{m^{2} + 1}

In view of the formula (5.2) for L(s, π_{p}), we find that |α_{π}(p, j)| ≤ p^{1/2−1/(m^{2}+1)} in the ramified case as well.

58 ZEÉV RUDNICK AND PETER SARNAK

As for the analogous improvement at infinity of (2.5), it is shown in [16] by a quite different method that for π_{∞} spherical, we have exactly as in the finite places

| Re μ_{π}(j)| ≤ \frac{1}{2} − \frac{1}{m^{2} + 1}  (5.21)

Acknowledgement: We would like to thank S. Miller for a careful reading of the manuscript and F. Shahidi for information concerning Rankin-Selberg L-functions.

References

1. L. Barthel and D. Ramakrishnan A nonvanishing result for twists of L-functions of GL(n), Duke Math. Jour. 74 (1994), 681–700.

2. H. Davenport Multiplicative Number Theory, second edition Springer-Verlag 1980.

3. F.J. Dyson Statistical Theory of the Energy Levels of Complex Systems, III, J. Math. Phys. 3 (1962), 166–175.

4. I.M. Gelfand and D. Kazhdan Representation of the group GL(n, K) where K is a local field, in Lie Groups and their Representations, ed. I.M. Gelfand, John Wiley and Sons, New York 1974, 95–118.

5. R. Godement and H. Jacquet Zeta Functions of Simple Algebras, Lecture Notes in Math. 260, Springer-Verlag 1972.

6. D.A. Hejhal On the Triple Correlation of Zeros of the Zeta Function, International Math. Res. Notices no. 7 (1994), 293–302.

7. H. Jacquet Principal L-Functions of the Linear Group, Proc. Symp. Pure Math. 33 vol. 2, 63–86, American Math. Soc., Providence 1979.

8. H. Jacquet, I.I. Piatetski-Shapiro and J.A. Shalika Rankin-Selberg Convolutions, Amer. Jour. of Math 105 (1983), 367–464.

9. H. Jacquet, I.I. Piatetski-Shapiro and J.A. Shalika Conducteur des représentations du groupe linéaire, Math. Ann. 256 (1981), 199–214.

10. H. Jacquet and J.A. Shalika On Euler products and the classification of automorphic representations I, Amer. Jour. of Math. 103 (1981), 499–558.

11. H. Jacquet and J.A. Shalika Rankin-Selberg Convolutions: Archimedean Theory, Israel Math. Conf. Proceedings 2 (1990), 125–208.

12. M. Kac Toeplitz matrices, translation kernels and a related problem in Probability Theory, Duke Math. Jour. 21 (1954), 501–509.

13. E. Landau Über die Anzahl der Gitterpunkte in gewisser Bereichen (Zweite Abhandlung) Gött. Nach. (1915) 209–243.

14. R.P. Langlands Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications, Lecture Notes in Math. 170, Springer, New York 1970, 18–86.

15. J.H. van Lint and R.M. Wilson A Course in Combinatorics, Cambridge University Press 1992.

16. W. Luo, Z. Rudnick and P. Sarnak On Selberg’s eigenvalue conjecture, to appear in Geometric and Func. Analysis.

17. W. Luo, Z. Rudnick and P. Sarnak On the “Ramanujan conjectures” for GL(m), in preparation.

18. M.L. Mehta Random Matrices, Second Edition, Academic Press 1991.

19. C. Mœglin and J.-L. Waldspurger Le Spectre Résiduel de GL(n), Ann. scient. Éc. Norm. Sup. 22 (1989), 605–674.

20. H.L. Montgomery The Pair Correlation of Zeros of the Zeta Function, Proc. Symp. Pure Math 24, 181–193, American Math. Soc., Providence 1973.

21. A.M. Odlyzko On the Distribution of Spacings Between Zeros of Zeta Functions, Math. of Comp. 48 (1987), 273–308.

22. A.M. Odlyzko The 10^{20} Zero of the Riemann Zeta Function and 70 Million of its neighbors, A.T. & T. preprint, 1989.

23. I.I. Piatetski-Shapiro Euler Subgroups, in Lie Groups and their Representations, ed. I.M. Gelfand, Halsted, New York 1975.

24. I.I. Piatetski-Shapiro Arithmetic Dirichlet Series: Conjectures, Proceedings of conference in honor of G. Freiman, Tel-Aviv.

25. B. Riemann Über die Anzahl der Primzahlen unter einer gegebenen Größe, Montasb. der Berliner Akad. (1858/60) 671–680, in Gessamelte Mathematische Werke 2nd edition Teubner, Leipzig 1982 no. VII.

26. Z. Rudnick and P. Sarnak The n-level correlations of zeros of the zeta function, C.R. Acad. Sci. Paris 319 (1994), 1027–1032 .

27. R. Rumely Numerical computations concerning the ERH, Math. Comp. 61 (1993), 415–440.

28. P. Sarnak Course notes, Princeton 1995.

29. A. Selberg Old and new conjectures and results about a class of Dirichlet series, Collected Papers vol. 2, Springer-Verlag, Berlin-Heidelberg-New York 1991.

30. J.-P. Serre Abelian ℓ-adic Representations and Elliptic Curves, W.A. Benjamin Inc. New York–Amsterdam 1968.

31. J.-P. Serre, letter to J.-M. Deshouillers (1981).

32. F. Shahidi On certain L-functions, Amer. Jour. of Math. 103 (1981), 297–355.

33. J. Shalika The Multiplicity One theorem for GL(n), Annals of Math. 100 (1974), 171–193.

34. F. Spitzer A Combinatorial Lemma and its Applications to Probability Theory, Trans. Amer. Math. Soc. 82 (1956), 323–339.

Raymond and Beverley Sackler School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel and Department of Mathematics, Princeton University, Princeton NJ 08544

Department of Mathematics, Princeton University, Princeton NJ 08544