There is a result of Soundararajan on the upper bound of the partial sums of the Möbius function assuming GRH [here](https://arxiv.org/abs/0705.0723). [Suger and Halupczok](https://projecteuclid.org/journals/functiones-et-approximatio-commentarii-mathematici/volume-48/issue-1/Partial-sums-of-the-M%C3%B6bius-function-in-arithmetic-progressions-assuming/10.7169/facm/2013.48.1.6.full) find an analogous bound for $\displaystyle \sum\_{\substack{n\leq x\\n \equiv a \bmod q}}\mu(n)$, also assuming GRH. However, without assuming GRH, what are some non-trivial upper bounds on $$\displaystyle \sum\_{\substack{n\leq x\\n \equiv a \bmod q}}\mu(n)?$$ It does not have to be the most optimal one (although I would appreciate being linked to such a bound). In fact, I don't even know of a non-trivial bound not assuming GRH on $\displaystyle \sum\_{n\leq x}\mu(n)$. Thanks!



Partial sums of the Möbius function on arithmetic progressions
A variant of the Siegel-Walfisz theorem states that there is a constant $c>0$ with the following property. For any $A>0$ and $q\leq(\log x)^A$, we have
$$\displaystyle \sum\_{\substack{n\leq x\\n \equiv a \bmod q}}\mu(n)\ll\_A x\exp\left(-c\sqrt{\log x}\right).$$
See Exercise 13 for Section 11.3 of Montgomery-Vaughan: Multiplicative number theory I.


