Generalized divisor functions in arithmetic progressions: I

TL;DR

This paper establishes new distribution results for the k-fold divisor function in arithmetic progressions beyond the square-root range of the modulus, using advanced techniques and assuming the Generalized Riemann Hypothesis for uniform error bounds.

Contribution

It extends distribution results for divisor functions to larger moduli and provides uniform error estimates under GRH, adapting methods from bounded prime gaps.

Findings

Proves distribution results for divisor functions in larger moduli ranges.

Achieves power-saving error terms assuming GRH, independent of k.

Provides effective estimates avoiding Siegel's theorem reliance.

Abstract

We prove some distribution results for the $k$-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length $X$ of the sum, with appropriate constrains and averaging on the moduli, saving a power of $X$ from the trivial bound. On assuming the Generalized Riemann Hypothesis, we obtain uniform power saving error terms that are independent of $k$. We follow and specialize Y.T. Zhang's method on bounded gaps between primes to our setting. Our arguments are essentially self-contained, with the exception on the use of Deligne's work on the Riemann Hypothesis for varieties over finite fields. In particular, we avoid the reliance on Siegel's theorem, leading to some effective estimates.