PIMS-BIRS TeamUp: Generalizations of the Prime Number Theorem (24rit201)

The Banff International Research Station will host the PIMS-BIRS TeamUp: "Generalizations of the Prime Number Theorem" workshop in Banff from July 28 to August 10, 2024.

The Prime Number Theorem for primes, along with its generalizations to primes in arithmetic progressions and primes in the Chebotarev density theorem, share similarities in their use of the zeros of associated zeta or $L$-functions to determine the size of the error term in approximating prime counting functions.

These similarities became evident in our projects on $\pi(x;q,a)$ (Lumley and Kadiri) and $\pi_C(x)$ (Das and Kadiri), where we studied these quantities explicitly. By combining information on zero density and zero-free regions to analyze infinite sums over the zeros, we observed that the summands are related to a common family of Bessel-type integral functions. These functions arise when the zeros in the sum lie off the 1/2-line of the Generalized Riemann Hypothesis. While bounds on these functions have been established for asymptotically large $x$, other cases have not been thoroughly investigated.

For large $x$, these functions are well understood. Our goal is to gain a comprehensive understanding of these functions by exploring cases with varying sizes of $x$ in relation to parameters such as moduli or the degree and discriminant of the field. This investigation will involve translating our case study to provide both numerically and asymptotically sharp estimates for the sums over the zeros.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta's Advanced Education and Technology.