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.