Sommes de Gál et applications

We evaluate the asymptotic size of various sums of Gál 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.

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