On the number of even values of an eta-quotient

Abstract

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient [Formula: see text], over any arithmetic progression. Namely, if [Formula: see text] denotes the number of even coefficients of [Formula: see text] in degrees [Formula: see text] (mod [Formula: see text]) such that [Formula: see text], then we show that [Formula: see text] is unbounded for [Formula: see text] large. Note that our result is very close to the best bound currently known even in the special case of the partition function [Formula: see text] (namely, [Formula: see text], proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre’s classical theorem on the number of even values of [Formula: see text], combined with a recent modular-form result by Cotron et al. on the lacunarity modulo 2 of certain eta-quotients. Interestingly, even in the case of [Formula: see text] first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre’s theorem.