On the quadratic variation of some sequences of fractional parts

Abstract

Let $a, b$ be real numbers such that $b > a \geq 0$, and let $c = b - a$. Let $x$ be a real number and $n$ a positive integer ($n \in \mathbb{N}^*$). The present study focuses on proving the following asymptotic formula
\[
\sum_{n \geq 1} \left( \left\{ \frac{x}{n+b} \right\} - \left\{ \frac{x}{n+a} \right\} \right)^2 = \frac{\zeta(3/2)}{\pi} \sqrt{cx} + O\left(c^{139/378} x^{61/126}\right),
\]
with $ c = b-a $, uniformly for $ x \geq 100 c^{-25/3} (1+b)^{29} $ generalization of the following result
\[
\sum_{n \geq 1} \left( \left\{ \frac{x}{n} \right\} - \left\{ \frac{x}{n+1} \right\} \right)^2 = \frac{\zeta(3/2)}{\pi} \sqrt{x} + O\left(x^{3/7}\right) \quad (x > 0),
\]
established in previous literature for the specific case $a=0$ and $b=1$.