  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.
