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

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.