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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)
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