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M. Balazard (2016)
Sur la variation totale de la suite des parties fractionnaires des quotients d'un nombre réel positif par les nombres entiers naturels consécutifsarXiv: Number Theory
(1923)
Corput – « Méthodes d’approximation dans le calcul du nombre des points à coordonnées entières
A. Wintner (1946)
Square root estimates of arithmetical sum functionsDuke Mathematical Journal, 13
AbstractLet b > a > 0. We prove the following asymptotic formula ∑n⩾0|{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 = b − a, uniformly for x ⩾ 40c−5(1 + b)27/2.
Communications in Mathematics – de Gruyter
Published: Dec 1, 2021
Keywords: Fractional part; Elementary methods; van der Corput estimates; 11N37
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