JournalspmVol. 64 , No. 2DOI 10.4171/pm/1779

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

  • Mireille Car

    Université Aix-Marseille III, France
Euler constants for the ring of <em>S</em>-integers of a function field cover

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Abstract

The Euler constant γ\gamma may be defined as the limit for nn tending to ++\infty, of the difference j=1n1jlogn\sum\limits_{j=1}^n\frac1{j}-\log n. Alternatively, it may be defined as the limit at 1 of the difference n=11js1s1\sum\limits_{n=1}^{\infty}\frac1{j^s}-\frac 1{s-1}, ss being a complex number in the half-plane (s)>1\Re(s)>1. Mertens theorem states that for xx real number tending to +\infty, px(11p)eγlogx\prod\limits_{p\leq x}(1-\frac{1}p)\sim \frac{e^{-\gamma}}{\log x}, the product being over prime numbers x\leq x. We prove analog results for the ring of SS-integers of a function field. However, in the function field case, the three approaches lead to different constants.