Finiteness of simple holomorphic eta quotients of a given weight

TL;DR

This paper proves that for any fixed weight, there are only finitely many holomorphic eta quotients that are not rescalings or products of simpler eta quotients, confirming a conjecture by Zagier and Mersmann.

Contribution

The paper provides a simplified proof of Zagier's conjecture, establishing finiteness of certain holomorphic eta quotients for each weight.

Findings

Finiteness of holomorphic eta quotients per weight

No rescaling or product of simpler eta quotients beyond trivial cases

Simplified proof of Zagier's conjecture

Abstract

We provide a simplified proof of Zagier's conjecture / Mersmann's theorem which states that of any particular weight, there are only finitely many holomorphic eta quotients, none of which is an integral rescaling of another eta quotient or a product of two holomorphic eta quotients other than 1 and itself.