A Montgomery-Hooley theorem for the k-fold divisor function

TL;DR

This paper extends the Montgomery-Hooley theorem to the k-fold divisor function, demonstrating an average bound for sums over arithmetic progressions for all k using a circle method approach.

Contribution

It generalizes previous results for k=2 to all k and employs a circle method to analyze the unsmoothed divisor sum problem.

Findings

Validates the expected bound for all k in an average sense

Generalizes prior work from k=2 to arbitrary k

Uses circle method to handle unsmoothed sums

Abstract

Let $d_k(n)$ denote the $k$-fold divisor function. For a wide range of large $q$ the expected bound $$\sum_{n\leq x\atop {n\equiv a(q)}}d_k(n)-\text { main term }\approx \sqrt {x/q}$$ is shown to be true in an average sense -- for all $k$. This generalises the work of Pongsriiam and Vaughan [15] who studied $k=2$, and answers the work of Rodgers and Soundararajan [17], who used the asymptotic large sieve to study a smoothed version of the problem. We use a circle method approach as developed by Goldston and Vaughan [7] to study the unsmoothed problem.