On the Hyperbolic Fractional Sum of the Divisor Function

Abstract

Let $\ddot{I}„(n)$ denote the classical divisor function. In this paper, we consider the hyperbolic fractional sum of the divisor function defined by $$T(x)=\sum _{n_1{n}_2⩽x}\ddot{I}„\left(\left[\frac{x}{n_1{n}_2}\right]\right)=\sum _{n⩽x}\ddot{I}„\left(\left[\frac{x}{n}\right]\right)\ddot{I}„(n),$$ 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 $\epsilon >0$, the error term in the asymptotic expansion of $T(x)$ is bounded by $O(x^{17/30+\epsilon })$. This result breaks the $4/7$-barrier which corresponds to the application of the classical divisor problem conjecture $1/4+\epsilon$.