Dérivabilité ponctuelle d’une intégrale liée aux fonctions de Bernoulli

Abstract:

Let $B (\vartheta )$ denote the first normalised Bernoulli function, and consider the series $f(\vartheta ):=\sum _{n\geqslant 1}B(n\vartheta )/n$. In a previous paper, we determined the set $E^*$ of those real numbers $\vartheta$ at which $f(\vartheta )$ converges. Let $B_2(\vartheta )$ designate the second Bernoulli function and let $\langle t\rangle$ denote the fractional part of the real number $t$. Put $F(\vartheta ):=\sum _{n\geqslant 1}{B_2(n\vartheta )/ 2n^2}=\pi ^2/72+\int _0^\vartheta f(t)\mathrm {d} t.$ It has been shown by Báez-Duarte, Balazard, Landreau and Saias (2005) that $F(\vartheta )$ and $\int _0^\infty \langle t\rangle \langle \vartheta t\rangle \mathrm {d} t/t^2$ have the same differentiability points. Moreover, using delicate functional equations, Balazard and Martin recently proved that the corresponding set is precisely $E:=E^*\smallsetminus \mathbb {Q}$ and that $F’(\vartheta )=f(\vartheta )$ whenever $\vartheta \in E$. We provide a short, direct proof of this last result, based on standard results from Diophantine approximation theory and uniform distribution theory.