Bounding sums of the Möbius function over arithmetic progressions

Let M(x) = ∑ 1≤n≤x μ(n) where μ is the Möbius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that M(x) = O(x1/2+ ) for all > 0. There has been much interest and progress in further bounding M(x) under the assumption of the Riemann Hypothesis. In 2009, Soundararajan established the current best bound of M(x) √ x exp ( (log x)(log log x) ) (setting c to 14, though this can be reduced). Halupczok and Suger recently applied Soundararajan’s method to bound more general sums of the Möbius function over arithmetic progressions, of the form M(x; q, a) = ∑ n≤x n≡a (mod q) μ(n). They were able to show that assuming the Generalized Riemann Hypothesis, M(x; q, a) satisfies M(x; q, a) √ x exp ( (log x)(log log x) ) for all q ≤ exp ( log 2 2 b(log x) 3/5(log log x)11/5c ) , with a such that (a, q) = 1, and > 0. In this paper, we improve Halupczok and Suger’s work to obtain the same bound for M(x; q, a) as Soundararajan’s bound for M(x) (with a 1/2 in the exponent of log x), with no size or divisibility restriction on the modulus q and residue a.