Habiba Kadiri, University of Lethbridge
An explicit version of Chebotarev’s Density Theorem.
This talk will first provide a (non-exhaustive) survey of explicit results on zero-free regions and zero densities of the Riemann zeta function and their relationship to error terms in the prime number theorem. This will be extended to Dirichlet L functions and Dedekind zeta functions, where new challenges arise with potential exceptional zeros. We will explore estimates for the error terms for prime counting functions across various contexts, with a specific attention to number fields. Chebotarev’s density theorem states that prime ideals are equidistributed among the conjugacy classes of the Galois group of any normal extension of number fields. An effective version of this theorem was first established by Lagarias and Odlyzko in 1977. In this article, we present an explicit refinement of their result. Key aspects of our approach include using the following: smoothing functions, recently established zero-free regions and zero-counting formula for zeros of the Dedekind zeta function, and sharp bounds for Bessel-type integrals. This is joint wok with Sourabh Das and Nathan Ng.
MC 2034