On sums over the Möbius function and discrepancy of fractions

We obtain quantitative upper bounds on partial sums of the Möbius function over semigroups of integers in an arithmetic progression. Exploiting the cancellation of such sums, we deduce upper bounds for the discrepancy of fractions in the unit interval [0,1] whose denominators satisfy the same restrictions. In particular, the uniform distribution and approximation of discrete weighted averages over such fractions are established as a consequence.