We show that for any positive integer N , there are only finitely many holomorphic eta quotients of level N , none of which is a product of two holomorphic eta quotients other than 1 and itself. This result is an analog of Zagier's conjecture/ Mersmann's theorem which states that: Of any given weight, there are only finitely many irreducible holomorphic eta quotients, none of which is an integral rescaling of another eta quotient. We construct such eta quotients for all cubefree levels. In particular, our construction demonstrates the existence of irreducible holomorphic eta quotients of arbitrarily large weights.
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.
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 q n 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.
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