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

### Mireille Car

Université Aix-Marseille III, France

## Abstract

The Euler constant $γ$ may be defined as the limit for $n$ tending to $+∞$, of the difference $j=1∑n j1 −gn$. Alternatively, it may be defined as the limit at 1 of the difference $n=1∑∞ j_{s}1 −s−11 $, $s$ being a complex number in the half-plane $ℜ(s)>1$. Mertens theorem states that for $x$ real number tending to +$∞$, $p≤x∏ (1−p1 )∼logxe_{−γ} $, the product being over prime numbers $≤x$. 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.

## Cite this article

Mireille Car, Euler constants for the ring of $S$-integers of a function field. Port. Math. 64 (2007), no. 2, pp. 127–142

DOI 10.4171/PM/1779