Research PaperNo Access

On the number of even values of an eta-quotient

Fabrizio Zanello

Department of Mathematical Sciences, Michigan Tech, Houghton, MI 49931, USA

https://doi.org/10.1142/S1793042123501105Cited by:0 (Source: Crossref)

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 $F$ , over any arithmetic progression. Namely, if $g_{a, b} (x)$ denotes the number of even coefficients of $F$ in degrees $n \equiv b$ (mod $a$ ) such that $n \leq x$ , then we show that $g_{a, b} (x) / \sqrt{x}$ is unbounded for $x$ large.

Note that our result is very close to the best bound currently known even in the special case of the partition function $p (n)$ (namely, $\sqrt{x} log log x$ , 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 $p (n)$ , 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 $p (n)$ 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.

Keywords:

AMSC: 11P83, 05A17, 11P82, 11F33

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