Hecke-type congruences for Andrews' spt-function modulo 16 and 32

TL;DR

This paper extends congruences for Andrews' spt-function modulo 16 and 32, demonstrating its normalized generating function as an eigenform under specific Hecke operators, revealing new modular properties.

Contribution

It introduces new Hecke-type congruences for the spt-function modulo 16 and 32, expanding understanding of its modular form behavior.

Findings

Normalized spt(n) generating function is an eigenform mod 32 for certain primes

Normalized spt(n) generating function is an eigenform mod 16 for other primes

Extends previous congruences to higher powers of 2 in the modulus

Abstract

Inspired by recent congruences by Andersen with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author for the function spt(n). We show that a normalized form of the generating function of spt(n) is an eigenform modulo 32 for the Hecke operators T(l^2) for primes l >= 5 with l congruent to 1, 11, 17, 19 (mod 24), and an eigenform modulo 16 for l congruent to? 13, 23 (mod 24).