Prime detecting quasi-modular forms in higher level

TL;DR

This paper extends the study of prime-detecting quasi-modular forms to higher levels, analyzing their structure and providing an analytic proof for the level one case, building on previous conjecture resolutions.

Contribution

It introduces a detailed description of the space of prime-detecting quasi-modular forms at higher levels and offers an analytic proof for the level one scenario.

Findings

Structured the space of prime-detecting quasi-modular forms at higher levels

Provided an analytic proof for the level one case

Extended previous conjecture resolution to higher levels

Abstract

In a previous work, the authors resolved a conjecture about the structure of prime-detecting quasi-modular forms by studying sign changes occurring in quasi-modular cusp forms. In this paper, we extend the considerations to prime-detecting quasi-modular forms of higher level, in particular describing the structure of the space of quasi-modular forms that detect primes in various arithmetic progressions. We also provide an ``analytic'' proof of the level one case.