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Dérivabilité ponctuelle d'une intégrale liée aux fonctions de Bernoulli


Authors: R. de la Bretèche and G. Tenenbaum
Journal: Proc. Amer. Math. Soc. 143 (2015), 4791-4796
MSC (2010): Primary 26A24, 26A27; Secondary 11J70, 11J71, 11K06, 11L07, 11M26
Published electronically: April 2, 2015
MathSciNet review: 3391036
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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)\textup {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 \textup {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^*\setminus \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.


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Additional Information

R. de la Bretèche
Affiliation: Institut de Mathématiques de Jussieu-PRG, Université Paris Diderot-Paris 7, Sorbonne Paris Cité, UMR 7586, Case 7012, F-75013 Paris, France
Email: regis.delabreteche@imj-prg.fr

G. Tenenbaum
Affiliation: Institut Élie Cartan, Université de Lorraine, BP 70239, F-54506 Vandœuvre-lès-Nancy Cedex, France
Email: gerald.tenenbaum@univ-lorraine.fr

DOI: https://doi.org/10.1090/S0002-9939-2015-12650-0
Keywords: Bernoulli functions, differentiability, Diophantine approximation, continued fractions, distribution modulo one, Erd\H{o}s-Tur\'an inequality, Davenport identities
Received by editor(s): December 11, 2013
Received by editor(s) in revised form: January 7, 2014, January 9, 2014, January 23, 2014, and July 23, 2014
Published electronically: April 2, 2015
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society