On the Hyperbolic Fractional Sum of the Divisor Function

TL;DR

This paper improves the asymptotic estimate for a hyperbolic fractional sum of the divisor function by establishing new bounds on exponential sums, surpassing previous barriers related to the divisor problem.

Contribution

It introduces new estimates for three-dimensional exponential sums, leading to a better error term in the asymptotic formula for the hyperbolic sum.

Findings

Error term bounded by O(x^{17/30+ε})

Breaks the 4/7 barrier in divisor sum estimates

Enhances understanding of divisor function sums in hyperbolic regions

Abstract

Let $\tau(n)$ denote the classical divisor function. In this paper, we consider the hyperbolic fractional sum of the divisor function defined by $$ T(x) = \sum_{n_1 n_2 \leqslant x} \tau\left( \left[ \frac{x}{n_1 n_2} \right] \right) = \sum_{n \leqslant x} \tau\left( \left[ \frac{x}{n} \right] \right) \tau(n), $$ where $[t]$ denotes the integral part of the real number $t$. By establishing new estimates for a class of three-dimensional exponential sums with constant perturbation, we obtain an improved asymptotic formula for $T(x)$. In particular, we show that for any $\varepsilon > 0$, the error term in the asymptotic expansion of $T(x)$ is bounded by $O(x^{17/30+\varepsilon})$. This result breaks the $4/7$-barrier which corresponds to the application of the classical divisor problem conjecture $1/4+\varepsilon$.