Bounding sums of the M\"obius function over arithmetic progressions

TL;DR

This paper improves bounds on sums of the Möbius function over arithmetic progressions, matching the best known bounds for the univariate case without restrictions on the modulus or residue, under the Generalized Riemann Hypothesis.

Contribution

It extends previous results to establish the same bounds for $M(x;q,a)$ as for $M(x)$, removing size and divisibility restrictions on the modulus and residue.

Findings

Bound on $M(x;q,a)$ matches Soundararajan's bound for $M(x)$

No restrictions on size or divisibility of $q$ and $a$

Results assume the Generalized Riemann Hypothesis

Abstract

Let $M(x)=\sum_{1\le n\le x}\mu(n)$ where $\mu$ is the M\"obius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that $M(x)=O(x^{1/2+\epsilon})$ for all $\epsilon>0$. There has been much interest and progress in further bounding $M(x)$ under the assumption of the Riemann Hypothesis. In 2009, Soundararajan established the current best bound of \[ M(x)\ll\sqrt{x}\exp\left((\log x)^{1/2}(\log\log x)^c\right) \] (setting $c$ to $14$, though this can be reduced). Halupczok and Suger recently applied Soundararajan's method to bound more general sums of the M\"obius function over arithmetic progressions, of the form \[ M(x;q,a)=\sum_{\substack{n\le x \\ n\equiv a\pmod{q}}}\mu(n). \] They were able to show that assuming the Generalized Riemann Hypothesis, $M(x;q,a)$ satisfies \[ M(x;q,a)\ll_{\epsilon}\sqrt{x}\exp\left((\log x)^{3/5}(\log\log x)^{16/5+\epsilon}\right) \] for all $q\le\exp\left(\frac{\log 2}2\lfloor(\log x)^{3/5}(\log\log x)^{11/5}\rfloor\right)$, with $a$ such that $(a,q)=1$, and $\epsilon>0$. In this paper, we improve Halupczok and Suger's work to obtain the same bound for $M(x;q,a)$ as Soundararajan's bound for $M(x)$ (with a $1/2$ in the exponent of $\log x$), with no size or divisibility restriction on the modulus $q$ and residue $a$.