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On the variation of certain fractional part sequences

Michel Balazard; Leila Benferhat; Mihoub Bouderbala — 2021

Communications in Mathematics

Let $b > a > 0$ . We prove the following asymptotic formula $\sum_{n ⩾ 0} | {x / (n + a)} - {x / (n + b)} | = \frac{2}{π} ζ (3 / 2) \sqrt{c x} + O (c^{2 / 9} x^{4 / 9}),$ with $c = b - a$ , uniformly for $x ⩾ 40 c^{- 5} {(1 + b)}^{27 / 2}$ .

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