Sur l'autocorrélation multiplicative de la fonction "partie fractionnaire" et une fonction définie par J. R. Wilton

Michel Balazard (1) , Bruno Martin (2)

1 IML - Institut de mathématiques de Luminy
2 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

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 Bretèche and Tenenbaum.

Mots clés: Fractional part; Autocorrelation; Continued fractions; Approximate functional equations

Domaines: Théorie des nombres [math.NT]

Michel Balazard, Bruno Martin. Sur l'autocorrélation multiplicative de la fonction "partie fractionnaire" et une fonction définie par J. R. Wilton. 2013. ⟨hal-00823899⟩