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Partial sums of the Möbius function in arithmetic progressions assuming GRH. (English) Zbl 1292.11107
Summary: We consider Mertens’ function in arithmetic progression, $M(x,q,a) := \sum{_{\substack{ n\leq x,\\ n\equiv a\bmod q}}} \mu(n).$ Assuming the generalized Riemann hypothesis (GRH), we show that the bound $M(x,q,a)\ll_\varepsilon \sqrt{x}\exp\left((\log x)^{3/5}(\log\log x)^{16/5 +\varepsilon}\right)$ holds uniform for all $$q\leq \exp(\tfrac{\log 2}{2}\lfloor (\log x)^{3/5}(\log\log x)^{11/5}\rfloor),$$ $$\gcd(a,q)=1$$ and all $$\varepsilon>0$$. The implicit constant depends only on $$\varepsilon$$. For the proof, a former method of K. Soundararajan is extended to $$L$$-series.

##### MSC:
 11N37 Asymptotic results on arithmetic functions 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)
##### Keywords:
Möbius function; Mertens function; GRH
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##### References:
 [1] D.A. Goldston, S.M. Gonek, A note on $$S(t)$$ and the zeros of the Riemann zeta-function , Bull. London Math Soc. vol. 39 (2007), no.3, 482-486. · Zbl 1127.11058 · doi:10.1112/blms/bdm032 [2] H. Iwaniec, E. Kowalski, Analytic number theory , American Math. Soc. Coll. Pub., vol. 53 (2004). · Zbl 1059.11001 [3] E. Landau, Über die Möbiussche Funktion , Rend. Circ. Mat. Palermo 48 (1924), 277-280. · JFM 50.0120.01 [4] H. Maier, H.L. Montgomery, The sum of the Möbius function , Bull. London Math Soc. 41 (2009), no. 2, 213-226. · Zbl 1241.11121 · doi:10.1112/blms/bdn119 [5] A. De Roton, M. Balazard, Notes de lecture de l’article,Partial sums of the Möbius function” de Kannan Soundararajan , arXiv.org, 2008, arXiv: · arxiv.org [6] A. Selberg, Lectures on sieves , Collected Papers, vol. 2 (1989), Springer, 65-247. [7] K. Soundararajan, Partial sums of the Möbius function , J. Reine Angew. Math. 631 (2009), 141-152. · Zbl 1184.11040 · doi:10.1515/CRELLE.2009.044 [8] E.C. Titchmarsh, A consequence of the Riemann hypothesis , J. London Math. Soc. 2 (1927), 247-254. · JFM 53.0312.02 [9] J.D. Vaaler, Some extremal functions in Fourier analysis , Bull. of the AMS 12 (1985), no. 2, 183-216. · Zbl 0575.42003 · doi:10.1090/S0273-0979-1985-15349-2
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