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

Ajit Singh and Rupam Barman

Vol. 320 (2022), No. 2, 425–436
Abstract

For a positive integer t, let bt(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 bt(n) modulo 2 for certain values of t. Further, they proposed some conjectures on self-similarities of bt(n) modulo 2 for certain values of t. For example, for a positive proportion of primes p, they conjectured that b3(n) satisfies

n=0b 3(2(pn + α))qn n=0b 3(2n)qpn(mod2),

where α 241(modp2), 0 < α < p2. In this paper, we prove their conjectures on b3(n) and b25(n). We also prove a self-similarity result for b21(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 Γ0(N), then for any positive integer M, a positive proportion of the primes p 1(modMN) has the property that f(z)Tp 0(modM).

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
Authors
Ajit Singh
Department of Mathematics
Indian Institute of Technology Guwahati
Assam
India
Rupam Barman
Department of Mathematics
Indian Institute of Technology Guwahati
Assam
India