On a conjecture about prime-detecting quasimodular forms

TL;DR

This paper proves a conjecture that classifies quasimodular forms which detect primes through their Fourier coefficients, demonstrating that these coefficients change sign infinitely often, thus linking prime detection to quasimodular forms.

Contribution

It confirms the conjecture by Craig, van Ittersum, and Ono, establishing the prime-detecting property of certain quasimodular forms and analyzing their Fourier coefficient behavior.

Findings

Fourier coefficients of these forms vanish exactly at primes

Fourier coefficients exhibit infinitely many sign changes

The conjecture on prime-detecting quasimodular forms is proven

Abstract

Motivated by weighted partition of $n$ that vanish if and only if $n$ is a prime, Craig, van Ittersum, and Ono conjecture a classification of quasimodular forms which detect primes in the sense that the $n$-th Fourier coefficient vanishes if and only if $n$ is a prime. In this paper, we prove this conjecture by showing that Fourier coefficients of quasimodular cusp forms exhibit infinitely many sign changes.