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# Portugaliae Mathematica

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*Portugaliae Mathematica*is restricted to the subscribers of the journal, who are encouraged to communicate their IP-address(es) to their agent or directly to the publisher at

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**Volume 64, Issue 2, 2007, pp. 127–142**

**DOI: 10.4171/PM/1779**

Published online: 2007-06-30

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

^{[1]}(1) Université Aix-Marseille III, France

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.

*Keywords: *Function field, Euler constant

Car Mireille: Euler constants for the ring of *S*-integers of a function field. *Port. Math.* 64 (2007), 127-142. doi: 10.4171/PM/1779