Residue-class distribution of arithmetic functions to varying moduli

The distribution of values of arithmetic functions in residue classes is a problem of significant interest in elementary, analytic and combinatorial number theory. Much work has been done studying this problem for fixed moduli. In this thesis, we extend many of the results in the literature for large classes of additive and multiplicative functions, so as to allow the modulus to vary within a wide range. In fact, we find essentially best possible analogues of the Siegel-Walfisz theorem (from prime number theory) for the joint distribution of families of such functions. Our primary tools are sieve methods and methods from the “anatomy of integers”, which we often use to detect certain “mixing” phenomena in multiplicative groups. Additionally, we use several ideas and machinery from classical analytic number theory, character sums, linear algebra over rings, as well as tools from arithmetic and algebraic geometry.