Number Theory Day

Date: October 27, 2022.

Venue: Euler Lecture Room, ICTP Leonardo Da Vinci Building, ICTP - Trieste.

Zoom registration link: https://zoom.us/meeting/register/tJYsdO2hqT0jGtYNXgFwuWnnmlh9bq2jWhpN

Program:

11:00 - 12:00: Micah Milinovich (Univ. of Mississippi, USA)

Title: Biases in the gaps between zeros of Dirichlet L-functions.

Abstract: We describe a family of Dirichlet L-functions that provably have unusual value distribution and experimentally have a significant and previously undetected bias in the distribution of gaps between their zeros. This has an arithmetic explanation that corresponds to the nonvanishing of a certain Gauss-type sum. We give a complete classification of the characters for when these sums are nonzero and count the number of corresponding characters. It turns out that this Gauss-type sum vanishes for 100% of primitive Dirichlet characters, so L-functions in our newly discovered family are rare (zero density set amongst primitive characters). If time allows, I will also describe some newly discovered experimental results concerning a "Chebyshev-type" bias in the gaps between the zeros of the Riemann zeta-function. This is joint work with Jonathan Bober and Zhenchao Ge.

13:30 - 14:30: Danylo Radchenko (Univ. of Lille, France)

Title: Dirichlet eigenvalues of regular polygons.

Abstract: For N>4 the first Dirichlet eigenvalue of a regular N-gon does not seem to have any closed form expression. However, it has been previously observed that it has an asymptotic expansion in powers of 1/N whose low order coefficients can be expressed in terms of special values of the Riemann zeta function at positive integers. It turns out that all coefficients of this expansion can be computed in closed form, but in general they involve more general multiple zeta values that (conjecturally) cannot be expressed in terms of the usual zeta values. I will discuss the proof of this result as well as some curious results and identities that arise in the proof. The talk is based on a joint work with D. Berghaus, B. Georgiev, and H. Monien.

15:00 - 16:00 Umberto Zannier (SNS Pisa, Italy)

Title: Finiteness theorems on elliptical billiards I: Poncelet games, elliptical billiards and relative Manin-Mumford.

Abstract: We shall recall the connection between elliptical billiards (a special case of Poncelet games) and certain elliptic families with sections. We shall show how fairly recent results of Manin-Mumford type imply finiteness for suitable patterns in the billiard. For example there are only finitely many possible shots for a player from a given position (outside a focus), which hit the opponent’s ball so that this falls eventually into a hole.

16:30 - 17:30: Pietro Corvaja (Univ. of Udine, Italy)

Title: Finiteness theorems on elliptical billiards II: Elliptical billiards and dynamical Mordell-Lang.

Abstract: We fit the finiteness results of the first talk into the frame of a general conjecture of dynamical type. This embraces further cases which sometimes can be proved by completely different arguments. We shall conclude with some distributional results of periodic orbits in elliptical billiards.