Sommes de Gál et applications

Abstract

We evaluate the asymptotic size of various sums of Gál type, in particular $$S(\mathcal{M}):=\sum _{m,n\in \mathcal{M}}\sqrt{\frac{(m,n)}{[m,n]}},$$ where $\mathcal{M}$ is a finite set of integers. Elaborating on methods recently developed by Bondarenko and Seip, we obtain an asymptotic formula for $$\log (\underset{|\mathcal{M}|=N}{\sup }S(\mathcal{M})/N)$$ 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.

arXiv commentin French. v2: corrected quote (p.3) from Soundararajan (2008), due to a misprint in the published version of this article. v3: corrected an inaccuracy with no consequence on the statements; v4, v5: minor typos and inaccuracies corrected; v6: corrected an inaccuracy in the proof of thm 1.6; v7: final, accepted version