Fractions de Bernoulli-Carlitz et op\'erateurs q-Zeta

TL;DR

This paper introduces a q-deformation of Dirichlet series as operators on formal power series, linking Bernoulli-Carlitz numbers to q-Riemann Zeta operators evaluated at negative integers, expanding the understanding of q-analogues in number theory.

Contribution

It presents a novel q-deformation of Dirichlet series as operators, connecting Bernoulli-Carlitz numbers to q-Riemann Zeta operators at negative integers.

Findings

Bernoulli-Carlitz numbers are images of certain polynomials under q-operators

Established a relation between Bernoulli-Carlitz numbers and q-Riemann Zeta operators

Introduced a new framework for q-deformations of Dirichlet series

Abstract

We introduce a q-deformation of Dirichlet series : for each s, an operator acting on formal power series in q without constant term. We relate Bernoulli-Carlitz numbers to the q-Riemann Zeta operators for negative integers, evaluated on some polynomials. ----- On introduit une d\'eformation des s\'eries de Dirichlet d'une variable complexe s, sous la forme d'un op\'erateur pour chaque nombre complexe s, agissant sur les s\'eries formelles en une variable q sans terme constant. On montre que les fractions de Bernoulli-Carlitz sont les images de certains polyn\^omes en q par les op\'erateurs associ\'es \`a la fonction ? de Riemann aux entiers n\'egatifs.