Sommes de G\'al et applications

TL;DR

This paper studies the asymptotic behavior of sums of Gál type over finite sets of integers, providing formulas and bounds that relate to extreme values of the Riemann zeta-function and Dirichlet L-functions.

Contribution

It develops asymptotic formulas for Gál sums and derives new bounds for extreme values of important number-theoretic functions.

Findings

Asymptotic formula for the supremum of Gál sums over sets of size N.

New lower bounds for localized extreme values of the Riemann zeta-function.

Bounds for extremal values of Dirichlet L-functions at s=1/2.

Abstract

We evaluate the asymptotic size of various sums of G\'al type, in particular $$S( \mathcal{M}):=\sum_{m,n\in\mathcal{M}} \sqrt{(m,n) \over [m,n]},$$ where $\mathcal{M}$ is a finite set of integers. Elaborating on methods recently developed by Bondarenko and Seip, we obtain an asymptotic formula for $$\log\Big( \sup_{|\mathcal{M}|= N}{S( \mathcal{M})/N}\Big)$$ and derive new lower bounds for localized extreme values of the Riemann zeta-function, for extremal values of some Dirichlet $L$-functions at $s=1/2$, and for large character sums.