On conjectures of Sato-Tate and Bruinier-Kohnen

TL;DR

This paper explores prime density relations, extends Bruinier-Kohnen conjecture results to include CM cases with improved density measures, and proves Sato-Tate equidistribution for CM modular forms with prime number theorem-like error bounds.

Contribution

It advances understanding of prime densities, generalizes Bruinier-Kohnen conjecture results to CM cases, and provides a complete proof of Sato-Tate equidistribution with explicit error terms.

Findings

Established links between prime and natural number subset densities.

Extended Bruinier-Kohnen conjecture results to include CM cases.

Proved Sato-Tate equidistribution for CM modular forms with explicit error bounds.

Abstract

This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato-Tate equidistribution for CM modular forms with an error term similar to that in the prime number theorem.