On the mean values of multiplicative functions on arithmetic progressions
Asymptotic formulas are given with remainder terms for the summation of complex-valued multiplicative functions on residue classes. The sum \(\sum_{n\leq x, n\equiv k (mod m)}f(n)\) is considered under the assumption that \[ \sum_{p\leq x, p\equiv j (mod m)}g(p)\log p=\alpha x+O(x\eta (x)) \] holds for all j coprime to m, and the remainder is estimated in terms of the function \(\eta\) (x); the results are too complicated to quote in detail.
sums of exponentially multiplicative functions
multiplicative functions on residue classes
scientific article; zbMATH DE number 4037941