Partial sums of the Möbius function in arithmetic progressions assuming GRH

Abstract

We consider Mertens' function in arithmetic progression, \[ M(x,q,a) := \sum{n\le x, n\equiv a mod q} \mu(n). \] Assuming the generalized Riemann hypothesis (GRH), we show that the bound \[ M(x,q,a)\ll_{\varepsilon} \sqrt{x}\exp{((\log x)^{3/5}(\log\log x)^{16/5 +\varepsilon})} \] holds uniform for all $q\le \exp(\frac{\log 2}{2}\lfloor (\log x)^{3/5}(\log\log x)^{11/5}\rfloor)$, $\gcd(a,q)=1$ and all $\varepsilon>0$. The implicit constant is depending only on $\varepsilon$. For the proof, a former method of K. Soundararajan is extended to $L$-series.