Functiones et Approximatio Commentarii Mathematici

Partial sums of the Möbius function in arithmetic progressions assuming GRH

Karin Halupczok and Benjamin Suger

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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.

Article information

Funct. Approx. Comment. Math., Volume 48, Number 1 (2013), 61-90.

First available in Project Euclid: 25 March 2013

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Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas

Möbius function Mertens' function GRH


Halupczok, Karin; Suger, Benjamin. Partial sums of the Möbius function in arithmetic progressions assuming GRH. Funct. Approx. Comment. Math. 48 (2013), no. 1, 61--90. doi:10.7169/facm/2013.48.1.6.

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