Euler constants for the ring of <em>S</em>-integers of a function field

Abstract

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.