The divisor function in arithmetic progressions modulo prime powers

TL;DR

This paper extends the understanding of the distribution of the divisor function in arithmetic progressions, specifically for prime power moduli, surpassing previous limitations for larger moduli when the modulus is a prime power with an odd prime.

Contribution

It introduces new methods to analyze the divisor function in arithmetic progressions modulo prime powers, surpassing the previous $x^{2/3}$ barrier for certain moduli.

Findings

Distribution of the divisor function in prime power progressions is better understood for larger moduli.

New techniques allow analysis beyond the $x^{2/3}$ barrier for prime power moduli.

Results apply to odd primes and fixed exponents $k \\ge 7$.

Abstract

We study the average value of the divisor function $\tau(n)$ for $n\le x$ with $n \equiv a \bmod q$. The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$. We show how to go past this barrier when $q=p^k$ for odd primes $p$ and any fixed integer $k\ge 7$.