Journal Details
Format
Journal
First Published
16 Jun 2010
Publication timeframe
2 times per year
Languages
English
Open Access

# Sur la variation de certaines suites de parties fractionnaires

###### Accepted: 29 Jul 2019
Journal Details
Format
Journal
First Published
16 Jun 2010
Publication timeframe
2 times per year
Languages
English

Let b > a > 0. We prove the following asymptotic formula

n0|{x/(n+a)}-{x/(n+b)}|=2πζ(3/2)cx+O(c2/9x4/9),\sum\limits_{n \geqslant 0} {\left| {\left\{ {x/\left( {n + a} \right)} \right\} - \left\{ {x/\left( {n + b} \right)} \right\}} \right| = {2 \over \pi }\zeta \left( {3/2} \right)\sqrt {cx} + O\left( {{c^{2/9}}{x^{4/9}}} \right),}

with c = ba, uniformly for x ⩾ 40c 5(1 + b)27/2.

#### MSC 2010

[1] M. Balazard: Sur la variation totale de la suite des parties fractionnaires des quotients d’un nombre réel positif par les nombres entiers naturels consécutifs. Mosc. J. Comb. Number Theory 7 (2017) 3–23.Search in Google Scholar

[2] J.G. van der Corput: Méthodes d’approximation dans le calcul du nombre des points à coordonnées entières. Enseign. Math. 23 (1923) 5–29.Search in Google Scholar

[3] J.G. van der Corput: Neue zahlentheoretische Abschätzungen. Math. Ann. 89 (1923) 215–254.Search in Google Scholar

[4] J.G. van der Corput: Zahlentheoretische Abschätzungen mit Anwendung auf Gitterpunktprobleme. Math. Z. 17 (1923) 250–259.Search in Google Scholar

[5] A. Wintner: Square root estimates of arithmetical sum functions. Duke Math. J. 13 (1946) 185–193.Search in Google Scholar

• #### A note on the volume of ∇-Einstein manifolds with skew-torsion

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