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

  • Mireille Car

    Université Aix-Marseille III, France

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.

Cite this article

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

DOI 10.4171/PM/1779