Zeta regularization of infinite products

The problem of regularizing infinite products is a long standing one in mathematical and theoretical physics, as well as in mathematics, of course. Several either formal or rigorous techniques have been introduced. One that obtained an undiscussed favor is the zeta regularization. The idea is simple to write down in a formal way (so as it as been used for a long times by physicists), and has got a rigorous mathematical formulation. This produced a huge number of interesting (and uninteresting) applications, with many useful (and non-useful) results. In this notes, we briefly recall the main features of the zeta regularization procedure, in particular following the approach introduced in our works. Given a sequence S = {an}∞n= 1 of non-vanishing complex numbers, arranged by increasing module, the formal product