Abstract: We say that a prime p is an Artin prime for g if g is a primitive root mod p. For appropriately chosen integers g and d, we present a conjecture for the asymptotic number of prime pairs (p,p+d) such that both p and p+d are Artin primes for g. Our model suggests that the distribution of Artin prime pairs, amongst the ordinary prime pairs, is largely governed by a Poisson binomial distribution. Time permitting, we moreover present a conjecture for the variance of Artin primes across short intervals of ordinary primes, obtained via similar heuristic methods (Joint work with Magdal?na Tinkov? and Mikul?? Zindulka).
Abstract: One of the most important objects in spectral geometry is the counting function for the eigenvalues λj for the Dirichlet Laplacian associated with planar domains. The simplest domains are squares, disks and ellipses. It is well-known that for each of these domains its eigenvalue counting function #{ λj ≤ μ2 } has an asymptotics containing two main terms a μ2 -b μ and a remainder of size o(μ). To improve the estimate of the remainder term had been one of the most attractive problems in spectral geometry for decades. I will introduce background briefly and explain how to transfer the above problem into problems of counting lattice points, to which tools from analysis and analytic number theory can be applied. I will mention our progresses for disks, annuli and balls in high dimensions, joint with Wolfgang Mueller, Weiwei Wang and Zuoqin Wang.
Abstract: The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes p< X such that p+h is prime for any non-zero even integer h.
While this conjecture remains wide open, Matomaki, Radziwill and Tao proved that it holds on average over h, improving on a previous result of Mikawa.
In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes.
We will describe some recent work in which we prove an asymptotic formula for the number of almost primes n=p1p2
Abstract: Cilleruelo conjectured that for any irreducible polynomial f with integer coefficients, and degree greater than one, the least common multiple of the values of f
at the first N integers satisfies log(lcm[f(1),...,f(N))~(deg(f)-1)Nlog(N), as N tends to infinity.
He proved this only for deg(f)=2. No example in higher degree is known. We study the analogue of this conjecture for function fields,
where we replace the integers by the ring of polynomials over a finite field. In that setting we are able to establish some instances of the conjecture for higher degrees.
The examples are all "special" polynomials f(X), which have the property that the bivariate polynomial f(X)-f(Y) factors into linear terms in the base field.
Abstract:
We will discuss global and local equidistribution of zeroes of power series
with coefficients r(n)/\sqrt{n!} where r is a sequence of complex-valued multipliers having binary correlations and no gaps in the spectrum.
We apply our approach to several examples of the sequence r of very different
origin, in particular various sequences of arithmetic origin such as the M?bius function where we see connections to Chowla's conjecture, random multiplicative
functions, and the function e(xn2) where the Diophantine nature of x plays a role.
The talk will be based on joint work with Alexander Borichev and Jacques Benatar
( arXiv:1908.09161, arXiv:2104.04812 )
May 6, 2021. Manuel Luethi (TAU). Random walks on homogeneous spaces, Spectral Gaps, and Khinchine's theorem on fractals cancelled
May 27, 2021. Jens Marklof (Bristol). cancelled
Abstract: Speiser in 1935 showed that the Riemann hypothesis is equivalent to the first derivative of the Riemann zeta function having no non-real zeros to the left of the critical line. This result shows a relation between the distribution of zeros of the Riemann zeta function and that of its derivative. Implications of the Riemann hypothesis to distribution of zeros of higher order derivatives are known but we are still yet to find an equivalence condition. Zeros of the derivatives of the Riemann zeta function in various setups were later studied by Spira, Berndt, and also Levinson and Montgomery. Among those results, a quantitative version of Speiser's 1935 result was proven by Levinson and Montgomery by showing that the number of non-real zeros of the Riemann zeta function does not differ much to those of its first derivative in 0 < Re(s) < 1/2 (the left-half of the critical strip). I expect that this is a formulation which is applicable to all derivatives. In this talk, I will introduce a few important results in this direction and new results obtained. Further, many results known for the Riemann zeta function have been generalized to Dirichlet L-functions and some are even extended to more general zeta and L-functions. I hope to give a brief introduction to what is known for Dirichlet L-functions and the difficulties in studying its higher order derivatives. I also hope to give an overview of common tools used in this study.
Abstract: Schinzel's Hypothesis states that every integer polynomial satisfying certain congruence conditions represents infinitely many primes. It is one of the main problems in analytic number theory but is completely open, except for polynomials of degree 1. We describe our recent proof of the Hypothesis for 100% of polynomials (ordered by size of coefficients). Furthermore, we give applications in Diophantine geometry. Joint work with Alexei Skorobogatov, preprint: https://arxiv.org/abs/2005.02998.
Abstract: The pair correlation function is a local measure of the randomness of a sequence. The behaviour of the pair correlation of sequences of the form ( {an α}) for almost every real number α where (an) is a sequence of integers is by now relatively well-understood. In particular, a connection to additive combinatorics was made by relating that behaviour to the additive energy of the sequence (an). Zeev Rudnick and Niclas Technau have recently started investigating the case of (an) being a sequence of real numbers. This talk is based on joint work with Christoph Aistleitner and Marc Munsch in which we pursue this line of research.
Abstract: We will discuss recent advances on the following two question: Let A(X) =Σ ±Xi be a random polynomial of degree n with coefficients taking the values -1, 1 independently each with probability 1/2.
Q1: What is the probability that A is irreducible as the degree goes to infinity
Q2: What is the typical Galois of A?
One believes that the answers are YES and THE FULL SYMMETRIC GROUP, respectively. These questions were studied extensively in recent years, and we will survey the tools developed to attack these problem and partial results.
Abstract: Euclidean lattice point counting problems, the classical example of which is the Gauss circle problem, are an important topic in classical analysis and have been the driving force behind much of the developments in the area of analytic number theory in the 20th century. While it is well known that homogeneous groups provide a natural setting to generalize many questions of Euclidean harmonic analysis, it was only recently that analogues of the Euclidean lattice point counting problem were considered for a certain family of 2-step nilpotent homogeneous groups. I will present the lattice point counting problem for Cygan-Koranyi norm balls on the Heisenberg groups, which is the analogue of the lattice point counting problem for Euclidean balls. I will describe recently obtained results relating to the distribution and moments of the error term on the Heisenberg groups, and discuss the similarities (and stark differences) between the Euclidean and Heisenberg case.
Abstract: Let a,b be multiplicatively independent positive integers. Bugeaud, Corvaja and Zannier (2003) proved that an-1, bn-1 have only a small common divisor, namely gcd(an-1, bn-1) < exp(ε n) for any fixed ε and sufficiently large n. Ailon and Rudnick (2004) were the first to consider the function field analogue and proved a much stronger result in this setting. These results triggered a floodgate of various extensions and generalisations, from the number case, to function fields in both zero and positive characteristics. For example, in the function field case besides powering there is another natural operation: iteration of functions. In this talk I will survey some of these results and their connections to some unlikely intersection problems for parametric curves. I will also discuss similar questions for linear recurrence sequences over function fields and pose some open questions.
Abstract: A natural way to test for the randomness of a sequence of points in R/Z is to consider its local statistics such as the k-level correlations and the nearest-neighbour gap distribution, and compare them to those of a sequence of uniform independent random points (Poisson statistics). In this talk I will describe recent results concerning two important examples of such sequences:
- The sequence {xn}, where in a joint work with Aistleitner, Baker and Technau we showed that for almost all x>1, all the correlations and hence the normalized gaps have a Poissonian limit distribution.
- The sequence {nx}, where in a joint work with Technau we showed Poissonian k-level correlations for almost all x sufficiently large (depending on k).
Abstract: The Littlewood conjecture in simultaneous approximation and the p-adic Littlewood conjecture are famous open problems in the intersection of number theory and dynamics. In a joint work with Faustin Adiceam and Fred Lunnon we show that an analogue of the p-adic Littlewood conjecture over F_3((1/t)) is false. The counterexample is given by the Laurent series whose coefficients are the regular paper folding sequence, and the method of proof is by reduction to the non vanishing of certain Hankel determinants. The proof is computer assisted and it uses substitution tilings of Z^2 and a generalisation of Dodgson's condensation algorithm for computing the determinant of any Hankel matrix.
Abstract: We explore the covariance of error terms coming from Weyl's conjecture regarding the number of Dirichlet eigenvalues up to size X. We also consider this problem in short intervals, i.e. the error term of the number of eigenvalues in the window [X, X+S] for some S(X). We look at these error terms for planar domains where the Dirichlet eigenvalues can be explicitly calculated. In these cases, the error term is closely related to the error term from the classical lattice points counting problem of expanding planar domains. We give a formula for the covariance of such error terms, for general planar domains. We also give a formula for the covariance of error terms in short intervals, for sufficiently large intervals. Going back to the Dirichlet eigenvalue problem, we give results regarding the covariance of the error terms in short intervals of 'generic' rectangles. We also explore a specific example, namely we compute the covariance between the error terms of an equilateral triangle and various rectangles.