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

摘要

We describe the points of diff erentiability of the multiplicativeautocorrelation function of the "fractional part" function. In connection withthis question, we study series involving the fi rst Bernoulli function, thearithmetical function "number of divisors", and the Gauss map from the theoryof continued fractions. A key role is played by a function defi ned in 1933 byJ. R. Wilton, similar to the Brjuno function of dynamical systems theory. Aunifying theme of our exposition is the use of functional equations involvingthe Gauss map, allowing us to reprove and re fine a theorem of Wilton, laBret\`eche and Tenenbaum.