A note on multiplicative functions on progressions to large moduli

TL;DR

This paper investigates the distribution of bounded multiplicative functions on arithmetic progressions to large moduli, establishing results for almost all primes within a certain large range.

Contribution

It extends understanding of multiplicative functions' distribution on progressions to larger moduli than previously known, approaching the square root of the range.

Findings

Well-distribution of multiplicative functions on progressions for large prime moduli

Results hold for moduli up to approximately $X^{1/2 + 1/78}$

Applicable to fixed residue classes, e.g., $a=1$

Abstract

Let $f : \mathbf{N} \rightarrow \mathbf{C}$ be a bounded multiplicative function. Let $a$ be a fixed integer (say $a = 1$). Then $f$ is well-distributed on the progression $n \equiv a \pmod{q} \subset \{1,\dots, X\}$, for almost all primes $q \in [Q,2Q]$, for $Q$ as large as $X^{\frac{1}{2} + \frac{1}{78} - o(1)}$.