On the number of even values of an eta-quotient

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arXiv:2210.09451v2 [math.CO] 5 May 2023

ON THE NUMBER OF EVEN VALUES OF AN ETA-QUOTIENT

FABRIZIO ZANELLO

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 ga,b(x) denotes the number of even coeﬃcients of Fin degrees n≡b

(mod a) such that n≤x, then we show that ga,b(x)/√xis unbounded for xlarge.

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, √xlog log x, proven by Bella¨ıche 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.

1. Introduction and preliminaries

The goal of this brief note is to present a general result on the longstanding problem of

estimating the number of even values of the Fourier coeﬃcients of arbitrary eta-quotients

(see below for the relevant deﬁnitions). In fact, we will do so over any arithmetic progression.

Namely, denoting by ga,b(x) the number of even coeﬃcients of an eta-quotient Fin degrees

n≡b(mod a) such that n≤x, in Theorem 2 we show that

ga,b(x)

√x

is unbounded for xlarge.

Our paper was originally motivated by the preprint [17], which asked whether a speciﬁc

eta-quotient assumes inﬁnitely many even, and inﬁnitely many odd values over any arithmetic

progression. (See Conjecture 2 of the arXiv version v3 of [17], which has since been updated

because of a mistake in one of the proofs, unrelated to our own paper. We thank the author

for informing us.) Our Theorem 2 positively answers the even part of that conjecture as a

2020 Mathematics Subject Classiﬁcation. Primary: 11P83; Secondary: 05A17, 11P82, 11F33.

Key words and phrases. Partition function; eta-quotient; binary q-series; modular form; parity of the

partition function.

2 FABRIZIO ZANELLO

very special case; for the odd part, see Question 5 at the end of this note, again in the much

broader framework of arbitrary eta-quotients.

The proof of Theorem 2 is substantially based upon, and generalizes, Serre’s classical

theorem [16] on the parity of the ordinary partition function p(n) (in fact, of a broader class

of functions) over any arithmetic progression, combined with a recent result by Cotron et al.

[5] on the lacunarity modulo 2 of eta-quotients satisfying a suitable technical assumption.

The use of the latter result, which implicitly requires the theory of modular forms, will

constitute the only non-elementary portion of our argument.

We note that even for the special case of p(n), Serre’s proof also required modular forms

in an essential fashion. In fact, while it is easy to see that the number of even values of the

partition function for n≤xhas order at least √x, no elementary proof is known to date

that this number grows faster than √x. At the end of this paper, we propose a problem,

phrased entirely in terms of quadratic representations, whose solution would ﬁnally yield a

modular form-free proof of Serre’s theorem for p(n).

We ﬁrst brieﬂy recall the main deﬁnitions. We refer the reader to, e.g., [8] and its references

for any unexplained terminology. Set fj=fj(q) = Qi≥1(1 −qji). Then an eta-quotient is a

quotient of the form

(1) F(q) = Qu

i=1 fri

αi

i=1 fsi

γi

for integers αiand γipositive and distinct, ri, si>0, and u, t ≥0. (Note that, for simplicity,

here we omit the extra factor of q(1/24)(Pαiri−Pγisi)that appears in some deﬁnitions of F,

since this factor is irrelevant for the asymptotic estimates of this paper.)

We say that F(q) = Pn≥0anqnis odd with density δif the number of odd coeﬃcients an

with n≤xis asymptotic to δx, for xlarge. Further, Fis lacunary modulo 2 if it is odd

with density zero (equivalently, if its number of odd coeﬃcients is o(x)).

One of the best-known instances of an eta-quotient (1) is arguably

n≥0

p(n)qn.

Understanding the parity of p(n) is a horrendously diﬃcult and truly fascinating problem,

which has historically attracted the interest of the best mathematical minds. A classical

conjecture by Parkin-Shanks [4, 13] predicts that p(n) is odd with density 1/2 (see [6, 7, 8]

for generalizations of this conjecture). However, the best bounds available today, obtained

after a number of incremental results, only guarantee that the even values of p(n) are of

order at least √xlog log x[3], and the odd values at least √x/ log log x[2].

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arXiv:2210.09451v2[math.CO]5May2023ONTHENUMBEROFEVENVALUESOFANETA-QUOTIENTFABRIZIOZANELLOAbstract.ThegoalofthisnoteistoprovideagenerallowerboundonthenumberofevenvaluesoftheFouriercoeﬃcientsofanarbitraryeta-quotientF,overanyarithmeticprogression.Namely,ifga,b(x)denotesthenumberofevencoeﬃcientsofFinde...

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