A note on the exponential sums of the localized divisor functions

TL;DR

This paper establishes an upper bound for exponential sums related to localized divisor functions, providing new estimates for the classical $k$-divisor function's exponential sums, which are useful in analytic number theory.

Contribution

It introduces a novel upper bound for exponential sums of localized divisor functions, enhancing understanding of their behavior and extending previous results for the $k$-divisor function.

Findings

Derived an explicit upper bound for exponential sums of localized divisor functions.

Provided estimates for the exponential sum of the classical $k$-divisor function.

Improved tools for analyzing divisor functions in analytic number theory.

Abstract

We prove an upper bound for the exponential sum associated to a localized $k-$divisor function, i.e., the counting function of the number of ways to write a positive integer $n$ as a product of $k\ge 2$ positive integers, each of them belonging to a specified interval. In particular, this gives an estimate for the exponential sum for the $k-$divisor function, $d_k(n)$.