Sur certaines \'equations fonctionnelles approch\'ees, li\'ees \`a la transformation de Gauss

TL;DR

This paper investigates functional equations related to the Gauss transformation in continued fractions, rederives a convergence criterion for a diophantine series, and provides new insights into its sum.

Contribution

It introduces new approximate functional equations linked to the Gauss transformation and extends classical results on diophantine series convergence and summation.

Findings

Reproves a convergence criterion for a diophantine series

Provides additional information about the series sum

Links functional equations to continued fraction transformations

Abstract

In the line of classical work by Hardy, Littlewood and Wilton, we study a class of functional equations involving the Gauss transformation from the theory of continued fractions. This allows us to reprove, among others, a convergence criterion for a diophantine series considered by Chowla, and to give additional information about the sum of this series.