Akash Singha Roy, University of Georgia
Residue-class distribution and mean values of multiplicative functions
The distribution of values of arithmetic functions in residue classes has been a problem of great interest in elementary and analytic number theory. The analogous question commonly studied for multiplicative functions is the distribution of their values in coprime residue classes. In work studying this problem for large classes of multiplicative functions, Narkiewicz obtained criteria deciding when a family of such functions is jointly uniformly distributed among the coprime residue classes to a fixed modulus. In the first part of this talk, we shall extend Narkiewicz's criteria to moduli that are allowed to vary in a wide range. Our results are essentially the best possible analogues of the Siegel-Walfisz theorem in this setting. One of the primary themes behind our arguments is the quantitative detection of a certain "mixing" (or ergodicity) phenomenon in multiplicative groups via methods belonging to the "anatomy of integers", but we also rely heavily on more classical analytic arguments, tools from arithmetic and algebraic geometry, and from linear algebra over rings.
In the second part of this talk, we shall gain a finer understanding of these distributions, such as the second-order behavior. This shall rely on extending some of the most powerful known estimates on mean values of multiplicative functions (precisely, the Landau-Selberg-Delange method) to a result that is much more uniform in certain important parameters. We will see several applications of this extended result in other interesting settings as well.
This talk is partially based on joint work with Prof. Paul Pollack.