Hyperbolic Summation for Fractional Sums

TL;DR

This paper develops an asymptotic evaluation of a specific double sum involving an arithmetic function and the floor function, utilizing advanced exponential sum estimates to extend understanding of fractional sums.

Contribution

It introduces a novel asymptotic formula for fractional sums involving two variables, applying three-dimensional exponential sum estimates to improve existing methods.

Findings

Derived asymptotic formulas for fractional sums

Applied exponential sum estimates to complex summations

Extended analytical techniques for sums involving floor functions

Abstract

Let $f(n)$ be an arithmetic function with $f(n) \ll n^\alpha$ for some $\alpha\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left( \left\lfloor \frac{x}{n_{1}n_{2}} \right\rfloor \right),$$ we use the estimation of three-dimensional exponential sums due to Robert and Sargos.