The Euler constant \gamma may be defined as the limit for n tending to +\infty , of the difference \sum\limits_{j=1}^n\frac1{j}-\log n . Alternatively, it may be defined as the limit at 1 of the difference \sum\limits_{n=1}^{\infty}\frac1{j^s}-\frac 1{s-1} , s being a complex number in the half-plane \Re(s)>1 . Mertens theorem states that for x real number tending to + \infty , \prod\limits_{p\leq x}(1-\frac{1}p)\sim \frac{e^{-\gamma}}{\log x} , the product being over prime numbers \leq x . We prove analog results for the ring of S -integers of a function field. However, in the function field case, the three approaches lead to different constants.

In odd dimension space, under a microlocal geometric condition, we give the rate of decay of the local energy for solutions of the wave equation on exterior domain, with localized nonlinear damping.