Sommes de Gál et applications

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