Small Gal sums and applications

In recent years, the question of maximizing GCD sums regained interest due to its firm
link with large values of L-functions, leading for instance to the breakthrough improvement of
Bondarenko and Seip concerning the maximum of the Riemann zeta  function on the critical line.
In this talk, we address the  counterpart problem of minimizing GCD sums and present several
results  with applications. We consider as well a related optimization question regarding the 
usual multiplicative energy of a subset of the first N integers. This is part of a joint work with
de la Bretèche and Tenenbaum.