On the Hyperbolic Fractional Sum of the Divisor Function

Abstract: Let $τ(n)$ denote the classical divisor function. In this paper, we consider the hyperbolic fractional sum of the divisor function defined by $$ T(x) = \sum_{n_1 n_2 \leqslant x} τ\left( \left[ \frac{x}{n_1 n_2} \right] \right) = \sum_{n \leqslant x} τ\left( \left[ \frac{x}{n} \right] \right) τ(n), $$ 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 $1/4+\varepsilon$.