Joint distribution in residue classes of families of multiplicative functions

Abstract: The distribution of values of arithmetic functions in residue classes has been a problem of great interest in elementary, analytic, and combinatorial number theory. In work studying this problem for large classes of multiplicative functions, Narkiewicz obtained general criteria deciding when a family of such functions is jointly uniformly distributed among the coprime residue classes to a fixed modulus. Using these criteria, he along with \' Sliwa, Rayner, Dobrowolski, Fomenko, and others, gave explicit results on the distribution of interesting multiplicative functions and their families in coprime residue classes.

In this talk, we shall give best possible extensions of Narkiewicz's criteria (and hence also of the other aforementioned results) to moduli that are allowed to vary in a wide range. This is motivated by the celebrated Siegel-Walfisz theorem on the distribution of primes in arithmetic progressions, and our results happen to be some of the best possible qualitative analogues of the Siegel-Walfisz theorem for the classes of multiplicative functions considered by Narkiewicz and others. Our arguments blend ideas from multiple subfields of number theory, as well as from linear algebra over rings, commutative algebra, and arithmetic and algebraic geometry. This talk is partly based on joint work with Paul Pollack.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)