Geodesic
On the variation of certain fractional part sequences
Communications in Mathematics, Tome 29 (2021) no. 3, pp. 407-430

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Let $b>a>0$. We prove the following asymptotic formula $$ \sum _{n\geqslant 0} \big \lvert \{x/(n+a)\}-\{x/(n+b)\}\big \rvert =\frac {2}{\pi }\zeta (3/2)\sqrt {cx}+O(c^{2/9}x^{4/9})\,, $$ with $c=b-a$, uniformly for $x \geqslant 40 c^{-5}(1+b)^{27/2}$.
Classification : 11N37
Keywords: Fractional part; Elementary methods; van der Corput estimates 
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     title = {On the variation of certain fractional part sequences},
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Balazard, Michel; Benferhat, Leila; Bouderbala, Mihoub. On the variation of certain fractional part sequences. Communications in Mathematics, Tome 29 (2021) no. 3, pp. 407-430. http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a6/