Rational exponential sums over the divisor function

TL;DR

This paper establishes nontrivial bounds for rational exponential sums over the divisor function by leveraging distribution properties of prime factors and applying advanced bounds from Bourgain, Korobov, and Shkredov.

Contribution

It introduces a novel approach combining distribution results and exponential sum bounds to analyze sums over the divisor function for general and prime moduli.

Findings

Derived bounds for exponential sums over the divisor function

Extended results to general and prime moduli cases

Connected divisor function sums with subgroup exponential sum bounds

Abstract

We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $\tau(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the distribution in residue classes of the function $\omega(n)$, counting the number of prime factors of $n$, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.