Partial sums of the M\"obius function in arithmetic progressions assuming GRH

TL;DR

This paper establishes a uniform upper bound for Mertens' function in arithmetic progressions under GRH, extending Soundararajan's method to L-series to improve understanding of the distribution of the Möbius function.

Contribution

It introduces a new uniform upper bound for Mertens' function in arithmetic progressions assuming GRH, by extending existing methods to L-series.

Findings

Provides a uniform upper bound for Mertens' function in arithmetic progressions.

Extends Soundararajan's method to L-series.

Assumes GRH for the results.

Abstract

We consider Mertens' function M(x,q,a) in arithmetic progression, Assuming the generalized Riemann hypothesis (GRH), we show an upper bound that is uniform for all moduli which are not too large. For the proof, a former method of K. Soundararajan is extended to L-series.