On the asymptotic behaviour of the square root of fractional part summatory function and generalization to the n-th root case

TL;DR

This paper derives an asymptotic expansion for the sum of fractional parts of square roots and generalizes it to n-th roots, exploring connections to special values of the Riemann zeta function.

Contribution

It introduces a new asymptotic expansion for the fractional part sum of square roots and extends the analysis to n-th roots with links to zeta function values.

Findings

Asymptotic expansion for sum of fractional parts of square roots

Generalization to n-th roots and their asymptotic behavior

Connection between fractional part sums and Riemann zeta function values

Abstract

We provide an asymptotic expansion for $\sum_{k=1}^n \left\{\sqrt{k}\right\}$. In the same spirit, we discuss the case of n-th root and it relation to special values of Riemman's zeta function.