Holomorphic eta quotients of weight 1/2

TL;DR

This paper provides a concise proof of Zagier's conjecture, classifying all holomorphic eta quotients of weight 1/2 as rescaled versions of a known list, and derives explicit coefficient formulas and level extensions.

Contribution

It offers a simplified proof of Zagier's conjecture and extends the understanding of holomorphic eta quotients of weight 1/2 with explicit formulas and level extensions.

Findings

All holomorphic eta quotients of weight 1/2 are rescaled from Zagier's list.

Explicit formulas for q-series coefficients of these eta quotients.

Extended levels for simple and irreducible holomorphic eta quotients.

Abstract

We give a short proof of Zagier's conjecture / Mersmann's theorem which states that each holomorphic eta quotient of weight 1/2 is an integral rescaling of some eta quotient from Zagier's list of fourteen primitive holomorphic eta quotients. In particular, given any holomorphic eta quotient $f$ of weight 1/2, this result enables us to provide a closed-form expression for the coefficient of qn in the $q$-series expansion of $f$, for all $n$. We also demonstrate another application of the above theorem in extending the levels of the simple (resp. irreducible) holomorphic eta quotients.