Nancy-Metz number theory seminar - Institut Élie Cartan de Lorraine

Upcoming presentations

On real zeros of the first derivative of quadratic Dirichlet $L$-functions

Catégorie d'évènement : Séminaire de Théorie des Nombres de Nancy-Metz Date/heure : 27 November 2025 14:30-15:30 Lieu : Salle de conférences Nancy Oratrice ou orateur : Kunjakanan Nath (IECL) Résumé :

One of the central topics in number theory is the study of $L$-functions and the distribution of their zeros. For example, the celebrated Prime Number Theorem is equivalent to the fact that the Riemann zeta function $\zeta(s)$ does not vanish on the line $\text{Re}(s)=1$. In this talk, we will focus on quadratic Dirichlet $L$-functions: in particular, the real zeros of the derivative of quadratic Dirichlet $L$-functions $L^\prime (s, \chi_d)$, where $d$ ranges over fundamental discriminants. Baker and Montgomery conjectured that there are $\asymp \log \log |d|$ real zeros of $L^\prime(s, \chi_d)$ in the interval $[1/2, 1]$ for almost all fundamental discriminants $d$. We will highlight some recent exciting progress that comes close to proving this conjecture and then outline the proof, which is based on ideas coming from analytic and probabilistic number theory. This is based on recent joint work with Youness Lamzouri.

Past presentations

On computing $L’/L(1,\chi)$ and related problems

Catégorie d'évènement : Séminaire de Théorie des Nombres de Nancy-Metz Date/heure : 11 March 2021 14:30-15:30 Lieu : Salle de séminaire de Théorie des Nombres virtuelle Oratrice ou orateur : Alessandro Languasco Résumé :

We first describe an efficient algorithm to compute
$L’/L(1,\chi)$, where $\chi$ is a non-principal Dirichlet character
mod q, and q is an odd prime. We then discuss
some results on the distribution of
$m_q := \min_{\chi\ne \chi_0} \vert L’/L(1,\chi) \vert $
and about the Euler-Kronecker constants for cyclotomic fields.

The distribution of random polynomials with multiplicative coefficients

Catégorie d'évènement : Analyse et théorie des nombres Date/heure : 18 February 2021 14:30-15:30 Lieu : Oratrice ou orateur : Brad Rodgers Résumé :

A classic paper of Salem and Zygmund investigates the distribution of trigonometric polynomials whose coefficients are chosen randomly (say +1 or -1 with equal probability) and independently. Salem and Zygmund characterized the typical distribution of such polynomials (gaussian) and the typical magnitude of their sup-norms (a degree N polynomial typically has sup-norm of size $\sqrt{N \log N}$ for large N). In this talk we will explore what happens when a weak dependence is introduced between coefficients of the polynomials; namely we consider polynomials with coefficients given by random multiplicative functions. We consider analogues of Salem and Zygmund’s results, exploring similarities and some differences.

Special attention will be given to a beautiful point-counting argument introduced by Vaughan and Wooley which ends up being useful.

This is joint work with Jacques Benatar and Alon Nishry.

Non-vanishing of cubic L-functions

Catégorie d'évènement : Séminaire de Théorie des Nombres de Nancy-Metz Date/heure : 21 January 2021 14:30-15:30 Lieu : Oratrice ou orateur : Alexandra Florea Résumé :

Résumé