Sur l'autocorr\'elation multiplicative de la fonction "partie fractionnaire" et une fonction d\'efinie par J. R. Wilton

TL;DR

This paper investigates the differentiability of the multiplicative autocorrelation of the fractional part function, connecting it with series involving Bernoulli functions, divisor functions, and the Gauss map, and refines Wilton's theorem using functional equations.

Contribution

It provides a detailed analysis of the autocorrelation function's differentiability and refines Wilton's theorem through functional equations involving the Gauss map.

Findings

Characterization of differentiability points of the autocorrelation function

Refinement of Wilton's theorem using functional equations

Connections between fractional parts, Bernoulli functions, and continued fractions

Abstract

We describe the points of diff erentiability of the multiplicative autocorrelation function of the "fractional part" function. In connection with this question, we study series involving the fi rst Bernoulli function, the arithmetical function "number of divisors", and the Gauss map from the theory of continued fractions. A key role is played by a function defi ned in 1933 by J. R. Wilton, similar to the Brjuno function of dynamical systems theory. A unifying theme of our exposition is the use of functional equations involving the Gauss map, allowing us to reprove and re fine a theorem of Wilton, la Bret\`eche and Tenenbaum.