Nancy-Metz number theory seminar

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.

In this talk, we will report on a joint work with Sanoli Gun,
where we study extreme values of $L$-functions attached to quadratic twists of Dirichlet characters.
We show that for any $\epsilon >0$ and Dirichlet character $F$ of odd conductor $q$, not necessarily a primitive form,
there exists at least $X^{1-\epsilon}$ fundamental discriminants $8d$ with $X< d \le 2X$ and $(d, 2q) =1$
such that $|L(1/2, F \otimes \chi_{8d})|$ takes large values.

Dans un travail récent, McNew, Pollack et Singha Roy obtiennent plusieurs résultats relatifs à la distribution du facteur premier médian des entiers lorsque ce dernier est défini en tenant compte de la multiplicité. En particulier, le comportement asymptotique des lois locales est étudié et fait apparaître une transition de phase qui n’est pas décrite. Dans cet exposé, nous présenterons une partie des améliorations et des résultats obtenus pour les lois locales et certaines applications.

In this talk, we will discuss diophantine approximations of irrational numbers by rational numbers, where the denominators are taken from certain interesting subsets of the positive integers. First, we will consider Diophantine approximations in which the denominators are drawn from the set of positive integers represented by a given positive definite integral binary quadratic form. Next, we will discuss Diophantine approximations where the denominators are restricted to the set of y-smooth (or friable) numbers for some given y > 0. Finally, we will outline some of the proofs.

Assume that $\omega_1, \ldots, \omega_n$ are i.i.d. uniform random points in $[0,1]$, which serve as the centers of shrinking intervals of given lengths $\ell_1 \ge \cdots \ge \ell_n$. The Dvoretzky covering problem asks for necessary and sufficient conditions on the sequence $(\ell_n)$ under which these random intervals cover $[0,1]$ infinitely often, almost surely. The problem was solved in 1972 by Shepp, and his work has since been generalized in several directions.

In this talk, I will discuss some deterministic analogues of Shepp’s result and their applications to the Littlewood Conjecture.

Nous développons une nouvelle technique pour minorer des sommes d’exponentielle de la forme $\sum f(n) e^{2i\pi\alpha n}$ pour tout $\alpha.$
Nous montrerons en particulier que la somme $\sum_{p\le x} e^{2i\pi\alpha p}$ est non bornée pour tout $\alpha,$ et plus précisément diverge au moins comme $x^{1/6-\varepsilon}$ pour une suite de $x$ tendant vers l’infini, uniformément en $\alpha.$

In this talk I will describe a way to implement the resonance method in problems of analytic number theory which are not necessarily multiplicative in nature.
This extension of the method not only produces improved extreme results wherever Dirichelt’s approximation theorem has been usually employed but it also highlights its connection to Bohr’s and Jessen’s proof of Kronecker’s approximation theorem.

En se laissant guider par l’exemple des ensembles de Sidon (ensembles de nombres dont les sommes de deux éléments sont uniques, très étudiés en combinatoire additive), je présenterai des résultats récents, en collaboration avec R. Riblet, où des techniques de théorie des ensembles permettent de construire des ensembles “grands” en certains sens (cardinalité, mesure ou dimension) tout en étant “épars” car évitant des configurations prescrites (pas de relation linéaire, ou ne contenant pas de parallélogramme, etc.). Des questions subtiles en lien avec l’axiome du choix seront évoquées.

Motivé par ses travaux et ceux de Behrend dans les années 30 concernant les ensembles primitifs d’entiers, Erdős conjectura en 1948 que si $\mathcal{A}$ est un ensemble dénombrable de réels $>1$, tel que $\limsup_{x\to +\infty} \frac{1}{\log x}\sum_{\alpha\leq x, \alpha\in \mathcal{A}}\frac{1}{\alpha} >0$, alors pour tout $\varepsilon>0$, il existe une infinité de triplets $(\alpha, \beta, n)\in \mathcal{A}^2\times \mathbb{N}$ tels que $\alpha\neq \beta$ et $|n\alpha-\beta|<\varepsilon.$ Très peu de temps avant sa mort en 1996, il avait offert 500$ pour la résolution de ce problème de nature diophantienne.

Dans cet exposé, je présenterai un travail récent, en collaboration avec Dimitris Koukoulopoulos et Jared Lichtman, où l’on démontre cette conjecture.

Random multiplicative functions are random models for arithmetic functions such as Dirichlet characters. Moments of sums involving random multiplicative functions are related to interesting counting problems, and understanding these counts can allow one to deduce the limiting distribution of the sums. Using this idea, Benatar, Nishry, and Rodgers showed that the limiting distribution of exponential sums with random multiplicative coefficients is Gaussian. However, they found that moments do not suffice if one wishes to understand the maximum size of these exponential sums. After introducing random multiplicative functions, we will discuss why this is the case, and show how one can obtain conjecturally sharp lower bounds for the maximum size of exponential sums with random multiplicative coefficients.

We consider some questions of arithmetic statistics for matrices of a given rank or fixed determinant or characteristic polynomial, whose entries are parametrised by arbitrary polynomials over the integers. In particular, some of our results improve a recent bound of V. Blomer and J. Li (2022) for counting matrices of given rank that are parametrised by monomials.

Joint works with Philipp Habegger, Ali Mohammadi and Igor Shparlinski.