Euler constants for the ring of $S$-integers of a function field

Abstract

The Euler constant $\gamma$ may be defined as the limit for $n$ tending to $+\infty$, of the difference $\sum _{j=1}^n\frac{1}{j}-\log n$. Alternatively, it may be defined as the limit at 1 of the difference $\sum _{n=1}^{\infty }\frac{1}{j^s}-\frac{1}{s-1}$, $s$ being a complex number in the half-plane $\Re (s)>1$. Mertens theorem states that for $x$ real number tending to +$\infty$, $\prod _{p\le x}(1-\frac{1}{p})\sim \frac{e^{-\gamma }}{\log x}$, the product being over prime numbers $\le x$. 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.