Translator Disclaimer
March 2013 Partial sums of the Möbius function in arithmetic progressions assuming GRH
Karin Halupczok, Benjamin Suger
Funct. Approx. Comment. Math. 48(1): 61-90 (March 2013). DOI: 10.7169/facm/2013.48.1.6

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.

Citation

Download Citation

Karin Halupczok. Benjamin Suger. "Partial sums of the Möbius function in arithmetic progressions assuming GRH." Funct. Approx. Comment. Math. 48 (1) 61 - 90, March 2013. https://doi.org/10.7169/facm/2013.48.1.6

Information

Published: March 2013
First available in Project Euclid: 25 March 2013

zbMATH: 1292.11107
MathSciNet: MR3086961
Digital Object Identifier: 10.7169/facm/2013.48.1.6

Subjects:
Primary: 11N37
Secondary: 11M06, 11M36

Rights: Copyright © 2013 Adam Mickiewicz University

JOURNAL ARTICLE
30 PAGES


SHARE
Vol.48 • No. 1 • March 2013
Back to Top