Proofs of some conjectures of Keith and Zanello on $t$-regular partition

Ajit Singh and Rupam Barman

Vol. 320 (2022), No. 2, 425–436

DOI: 10.2140/pjm.2022.320.425

Abstract

For a positive integer $t$, let $b_t(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results for $b_t(n)$ modulo $2$ for certain values of $t$. Further, they proposed some conjectures on self-similarities of $b_t(n)$ modulo $2$ for certain values of $t$. For example, for a positive proportion of primes $p$, they conjectured that $b_3(n)$ satisfies

$$\sum _{n=0}^{\infty }{b}_3(2(pn+\alpha ))q^n\equiv \sum _{n=0}^{\infty }{b}_3(2n)q^{pn}(mod2),$$

where $\alpha \equiv -24^{-1}(\mathrm{mod}<!--FUNCTION\; APPLICATION-->p^2)$, $0<\alpha <p^2$. In this paper, we prove their conjectures on $b_3(n)$ and $b_{25}(n)$. We also prove a self-similarity result for $b_{21}(n)$ modulo $2$. The proofs use a result of Serre which says that if $f$ is an integer weight cusp form on the congruence subgroup ${\Gamma }_0(N)$, then for any positive integer $M$, a positive proportion of the primes $p\equiv -1(\mathrm{mod}<!--FUNCTION\; APPLICATION-->M“N)$ has the property that $f(z)\mid T_p\equiv 0(\mathrm{mod}<!--FUNCTION\; APPLICATION-->M)$.

Keywords

$t$-regular partitions, eta-quotients, modular forms

Mathematical Subject Classification

Primary: 11P83, 11F11

Milestones

Received: 1 February 2022

Accepted: 14 September 2022

Published: 15 February 2023