THE n-LEVEL CORRELATIONS OF ZEROS OF THE ZETA FUNCTION

ZEÉV RUDNICK AND PETER SARNAK

To Paul Cohen on the occasion of his 60-th birthday

Abstract:

We define the n-level correlation sums for the normalized zeros of Riemann’s zeta function. A special case of these is the pair correlation function investigated by H. Montgomery. If one assumes the Riemann Hypothesis, then these sums measure the n-level correlations of the imaginary parts of the zeros, a statistic introduced by F. Dyson. It is shown that for a restricted class of test functions the n-level correlations follow the distribution predicted by the Gaussian Unitary Ensemble of random matrix theory. More generally, we show that the same is true universally for the principal L-functions attached to cuspidal automorphic representations of GL(m) over the rationals.

Supported by NSF grants DMS-9102082 and DMS-9400163.

2 ZEÉV RUDNICK AND PETER SARNAK

1. The Riemann zeta function

This note is concerned with the distribution of the zeros of the Riemann zeta function ζ(s). Let ρ_{j} = 1/2+iγ_{j}, j = ±1, ±2, … denote the non-trivial zeros of ζ(s). To motivate the formulation of our results we begin by assuming the Riemann Hypothesis (RH): γ_{j} ∈ R. We order the imaginary parts by γ_{1} < γ_{2} ≤ . . .. Their number is asymptotically

N(T) := #\{j ≥ 1 : γ_{j} ≤ T \} ∼ \frac{1}{2π} T log T,  T → ∞ (1.1)

It follows that the numbers γ̃_{j} := (γ_{j} log γ_{j})/2π have mean spacing unity. We investigate the statistics governing the distribution of the γ̃_{j}’s.

Following the original calculation by Montgomery [11] of the pair correlation (assuming RH and for a restricted class of test functions) and the extensive numerical investigations by Odlyzko [13], it is by now well accepted that the distribution of consecutive spacings follow the Gaussian Unitary Ensemble of Random Matrix Theory. That is, if we define the normalized spacings δ_{j} := γ̃_{j+1} − γ̃_{j}, then for any localized f on (0, ∞)

\frac{1}{N} ∑_{n=1}^{N} f(δ_{n}) → ∫_{0}^{∞} f(x)P (x)dx

where P (x)dx is the distribution of normalized spacings of large random Hermitian matrices (GUE distribution). The latter distribution was obtained by Gaudin and Mehta [10] and is given by P (s) = d^{2}E/ds^{2} 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) = (sin π(ξ − η)s/2)/π(ξ − η).

The consecutive level spacing distribution is determined by the nlevel correlation functions R_{n} which are defined as follows: Let B_{N} = {γ̃_{1} ≤ ⋯ ≤ γ̃_{N}}. The n-level correlations measure the distribution of the differences of n elements of B_{N}, 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)

An equivalent but 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 (i.e. faster than |x|^{−A} for any A > 0).

Define

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}} and in view of TF 1 this is well defined. TF 2 asserts that f is a function of the successive differences and hence we recover what (1.2) seeks to measure. 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. Schwartz functions). Here L = 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.

In [2], Dyson determined the 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 ∈ Z^{n} and x_{i} ≠ x_{j} for all i ≠ j.

It is important to 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 can be used to predict their asymptotic behaviour. Our first result proves that this prediction is correct for a restricted class of f’s.

Theorem 1.1. Let f satisfy TF 1,2,3 and in addition assume that f̂(ξ) is supported in ∑_{j} |ξ_{j}| < 2. Let g ∈ C_{c}^{∞}(R) and h(r) = ∫_{−∞}^{∞} g(u)e^{iru}du (so that h and f are entire). Then as T → ∞,

R_{n}(T, f, h) ∼ \frac{1}{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}

4 ZEÉV RUDNICK AND PETER SARNAK

where δ(x) is the Dirac mass at 0.

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

Theorem 1.2. Assume RH and that the support of f̂(ξ) is contained in ∑_{j} |ξ_{j}| < 2. Then as N → ∞

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

Remarks:

1. As can be seen from the next Section, the condition ∑_{j} |ξ_{j}| < 2 is “natural” in the sense that it is exactly the region in which the “multi-diagonals” give the main term in the asymptotics. Beyond this region the off-diagonal terms contribute equally. One conjectures [11] that Theorem 1.2 holds without the support restriction on f̂ - that is to say that the n-level correlations are those of GUE. E. Bogomolny and J. Keating have informed us that they have a heuristic proof of this (work in progress).

2. As mentioned before, Theorem 1.2 for n = 2 is due to Montgomery. Hejhal [5] has recently established Theorem 1.2 for the case n = 3 (in somewhat different form). Our approach is different to the one taken by these two authors. Firstly it directly establishes Theorem 1.1, which allows us to avoid appealing to the Riemann Hypothesis, and secondly it considerably facilitates the computations for general n.

2. Outline of the proofs

We give a brief outline of the main steps in proving Theorems 1.1 and 1.2. We use a smooth version of Riemann’s Explicit Formula [14]: If g ∈ C_{c}^{∞}(R) and h(r) = ∫_{−∞}^{∞} g(u)e^{iru}du then

∑_{γ} h(γ) = h(\frac{i}{2}) + h(−\frac{i}{2})

+ \frac{1}{2π} ∫_{−∞}^{∞} h(r) (\frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + ir) + \frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} − ir)) dr

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

(2.1) where Λ(n) = log p, if n = p^{k} is a prime power, and is 0 otherwise, and Γ_{R}(s) = π^{−s/2}Γ(\frac{s}{2}) is the Gamma factor appearing in the functional equation of ζ(s).

Let

C_{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_{1}}) (2.2)

(note that here we are not assuming that the indices j_{k} are distinct). Then for f satisfying TF 1,2,3 and f̂ of compact support

C_{n}(T, f, h) = ∫_{R^{n}} ∏_{k=1}^{n} \{∑_{j_{k}} h(\frac{γ_{j_{k}}}{T})e^{−iLγ_{j_{k}} ξ_{k}} \} dμ(ξ)  (2.3)

where

dμ(ξ) = Φ(ξ)δ(ξ_{1} + ⋯ + ξ_{n})dξ

is the Fourier transform of f. Applying the Explicit Formula (2.1) we have

\begin{aligned}C_{n}(T, f, h) & = ∫_{R^{n}} ∏_{k=1}^{n} (T g_{T} (T Lξ_{k}) + T S^{+}(ξ_{k}) + T S^{−}(ξ_{k}) \\ & +h(−\frac{i}{2T})T^{−ξ_{k}/2} + h(\frac{i}{2T})T^{ξ_{k}/2}) dμ(ξ) \end{aligned} (2.4)

where

g_{T} (x) = \frac{1}{2π} ∫_{−∞}^{∞} h(r) (\frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} + irT) + \frac{Γ′_{R}}{Γ_{R}} (\frac{1}{2} − irT)) e^{−irx}dr
S^{±}(ξ) = − ∑_{n=1}^{∞} \frac{Λ(n)}{\sqrt{n}}g(T (Lξ ± log n)) (2.5)

We multiply out the product in (2.4) and study the asymptotics as T → ∞ of the individual terms under the assumption that ∑_{j} |ξ_{j}| < 2. All terms of lower order than T log T are dropped (so for example this is the case for the full product involving h(±i/2T)T^{±ξ/2}). A typical term looks like

C_{r,s}(T) = ∑_{n} \frac{Λ(n_{1}) … Λ(n_{r+s})}{\sqrt{n_{1} … n_{r+s}}}A_{r,s}(n, T)  (2.6)

where

A_{r,s}(n, T) = T^{n} ∫_{R^{n}} ∏_{j=1}^{r} g(T(Lξ_{j} + log n_{j})) ∏_{j=r+1}^{r+s} g(T(Lξ_{j} − log n_{j})) ∏_{j>r+s} g_{T}(TLξ_{j})dμ(ξ) (2.7)

6 ZEÉV RUDNICK AND PETER SARNAK

Using the fact that g is of compact support and ∑_{j} |ξ_{j}| < 2, one shows that the main contributions to the sum in (2.6) are from n_{1}·· · ··n_{r+s} ≪ T^{2−δ}, δ = δ(f) > 0 and in addition n_{1} · · · · · n_{r} = n_{r+1} · · · · · n_{r+s}. A further analysis leads to

C_{r,s}(T) = \frac{1}{2π} ∫_{−∞}^{∞} h(r)^{n}dr\frac{T}{L^{r+s−1}} ∑_{\begin{matrix}n_{1}·····n_{r}=n_{r+1}·····n_{r+s} \\ n_{j}≪T \end{matrix}} \frac{Λ(n_{1}) … Λ(n_{r+s})}{n_{1} … n_{r}}

· Φ(−\frac{log n_{1}}{L} , …, −\frac{log n_{r}}{L} , \frac{log n_{r+1}}{L} , …, \frac{log n_{r+s}}{L} , 0, …, 0) + O(T) (2.8)

Since Λ is supported on prime powers, the above leads to the condition that n_{j} are prime (for the main term) and so there must be a pairing of primes in the relation (r = s)

p_{1} · · · · · p_{r} = p_{r+1} · · · · · p_{r+s}  (2.9)

This leads to C_{r,s}(T) = O(T) if r ≠ s, and

C_{r,r}(T) = \frac{T}{2πL^{2r−1}} ∫_{−∞}^{∞} h(r)^{n}dr ∑_{p_{1},...,p_{r}≪T} ∑_{σ∈S_{r}} \frac{log^{2} p_{1} … log^{2} p_{r}}{p_{1} … p_{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) (2.10)

where S_{r} is the symmetric group on r letters. To (2.10) we may apply the Prime Number Theorem and summation by parts to obtain

C_{n}(T, f, h) = \frac{T L}{2π} ∫_{−∞}^{∞} h(r)^{n}dr ∫_{R^{n}} C\underline{O}(v)Φ(v)dv + O(T) (2.11)

where

∫_{R^{n}} C\underline{O}(v)Φ(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} (2.12)

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 e_{i,j} = e_{i} − e_{j}, e_{i} = (0, …, 1, …, 0) are the standard basis vectors.

To prove Theorem 1.1 one must now relate C_{n}(T, f, h) which is a sum over all indices, to R_{n}(T, f, h) which is a sum over distinct ones. To this end we need some combinatorial sieving. The crux is to establish certain identities between the distribution C\underline{O}(v) and certain related distributions and the GUE determinant (1.5).

Let \underline{F} = [F_{1}, …, F_{ν}] be a set partition of \{1, …, n\}. For instance, we set \underline{O} = [\{1\}, …, \{n\}]. We define C\underline{F} (v) to be the distribution obtained from C\underline{O}(v) when the variables v_{l}, l ∈ F_{j}, are set equal to each other. Then

C\underline{F} (v) = δ\underline{F} (v) + ∑_{r=1}^{[n/2]} ∑ ∏_{t=1}^{r} δ_{F_{i(t)}∪F_{j(t)}}(v)| ∑_{l∈F_{i(t)}} v_{l}| ∏_{k≠i(a),j(b)} δ_{F_{k}}(v) (2.13)

the sum being over all possible pairs of subsets (F_{i(t)}, F_{j(t)}), no repetitions allowed. The δ-functions are defined by δ_{S}(v) = δ(∑_{l∈S} v_{l}) for S ⊂ \{1, …, n\} and for a set partition \underline{F} as above, δ\underline{F} (v) = ∏_{j=1}^{ν} δ_{F_{j}}(v).

The distribution R\underline{O}(v) that dictates the asymptotics of R_{n}(T, f, h) may be expressed via Möbius inversion as

R\underline{O}(v) = ∑_{\underline{F}} μ(\underline{O}, \underline{F})C\underline{F} (v)  (2.14)

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

The key identity is the following:

Proposition 2.1. For ∑_{j} |v_{j}| < 2, the Fourier transform Ŵ_{n}(v) equals R\underline{O}(v).

Proposition 2.1 in turn is based on the following identity between piecewise linear functions. It couples the pairing structure (2.9) (i.e. unique factorization) which is manifest in (2.13) and the cycle structure of the determinant defining W_{n}.

Proposition 2.2. Let v ∈ R^{n} with ∑_{j} v_{j} = 0. Then

\begin{matrix}∑_{[F,F^{c}]}(|F | − 1)!(|F^{c}| − 1)!| ∑_{l∈F} v_{l}| \\ = ∑_{\{θ\}} (max\{v_{θ(1)}, v_{θ(1)} + v_{θ(2)}, … \} − min\{v_{θ(1)}, v_{θ(1)} + v_{θ(2)}, … \}) \end{matrix}

where the first sum is over all partitions of {1, …, n} into two subsets, and the second sum is over all orderings θ = (θ(1), …, θ(n)) of {1, …, n} modulo rotations.

The proof of Proposition 2.2 is based on Spitzer’s combinatorial method [17]. Finally, the passage from Theorem 1.1 to Theorem 1.2 is a straight-forward approximation argument.

8 ZEÉV RUDNICK AND PETER SARNAK

3. General L-functions

Beginning with Dirichlet L-functions L(s, χ), we observe that the analysis and proofs above apply equally well to these. Hence the analogues of Theorems 1.1 and 1.2 may be proven for the zeros of such L-functions. Note however that if L(s) = ζ(s)L(s, χ) then we no can longer expect its zeros to follow GUE statistics. The reason is that for non-trivial χ, the zeros of ζ(s) and L(s, χ) “do not know about each other”, i.e. are statistically independent; for correlations this may be proved by the methods of Section 2. Indeed the distribution of zeros of such L(s) should be the superposition of two independent GUE’s, which is quite different to the original GUE distribution. In view of this feature we consider only primitive L-functions, that is ones which cannot be factored. Precisely, following Godement-Jacquet, we let L(s, π) be the principal L-function associated to an irreducible cuspidal automorphic representation of GL(m)/Q (see [4], [6]). These do not factor into products of other such L-functions (this is the reason for working over Q) and it follows from the general conjectures of Langlands [9] [1]

[3] that all known L-functions of number theory should be products of these. We call such an L-function a primitive principal L-function of degree m ≥ 1. The Dirichlet L-functions and ζ(s) are the only ones of degree one. The degree affects the density of zeros (1.1), which gets multiplied by m. The analogues of Theorems 1.1 and 1.2 can be proven for zeros of primitive principal L-functions, with some modifications. Firstly, we need to make a technical hypothesis (see (3.4) below) which at this point we can establish for many cases, including all those with m < 4. Secondly, the restriction on the support of f̂ is replaced by ∑ |ξ_{j}| < 2/m. If m ≥ 2, heuristic arguments show that the multidiagonals (2.7), (2.8) should still be the dominant term in the region 2/m ≤ ∑ |ξ_{j}| < 2, that is to say it should be the same as for m = 1. In fact we conjecture that in all these cases no restriction on the support of f̂ is needed at all. That is to say, zeros of primitive principal L-functions always follow the universal statistics of GUE. While such a universality is very pleasing it is also somewhat surprising. The reason being that if we write

\frac{L′}{L} (s, π) = − ∑_{n=1}^{∞} \frac{Λ(n)a_{π}(n)}{n^{s}}  (3.1)

then the distribution of {a_{π}(p)} for p prime depends on π. For instance in the case m = 2 they should follow either the Sato-Tate or uniform distribution [16], while for m > 2 there is an increasing number of possible distributions. What causes universality for the zeros is the pairing structure (2.9), (2.10). These show that it is the asymptotics of

σ_{π}(x) = ∑_{n≤x} \frac{|Λ(n)a_{π}(n)|^{2}}{n}  (3.2)

that are crucial. The analytic properties of the Rankin-Selberg Lfunction L(s, π ⊗ π̃) due to Jacquet, Piatetski-Shapiro and Shalika [7],

[8] are used to show that

σ_{π}(x) ∼ \frac{1}{2}(log x)^{2}  (3.3)

(this is independent of π!) granting the following technical hypothesis: For fixed k ≥ 2

∑_{p} \frac{|a_{π}(p^{k}) log p|^{2}}{p^{k}} < ∞  (3.4)

The general “Ramanujan conjectures” a_{π}(p^{k}) = O(1) are clearly more than enough to imply (3.4).

As mentioned above, the proof in general is conceptually the same as for ζ(s), the technical details however being more formidable. These and complete proofs of Theorems 1.1 and 1.2 will appear in the forthcoming article [15].

References

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

3. S. Gelbart An Elementary Introduction to the Langlands Program, Bull. Amer. Math. Soc. 10 (1984), 177–219.

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

5. D.A. Hejhal On the Triple Correlation of Zeros of the Zeta Function, preprint.

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

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

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

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

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

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

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

14. 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.

15. Z. Rudnick and P. Sarnak Zeros of Principal L-Functions and Random Matrix Theory, in preparation.

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

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

Department of Mathematics, Princeton University, Princeton NJ 08544