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Hecke Relations for Eta Multipliers and Congruences of Higher-Order Smallest Parts Functions

Clayton Williams

Abstract

We derive identities from Hecke operators acting on a family of Eisenstein-eta quotients, yielding congruences for their coefficients modulo powers of primes. As an application we derive systematic congruences for several higher-order smallest parts functions modulo prime powers, resolving a question of Garvan for these cases. We also relate moments of cranks and ranks to the partition function modulo prime powers. Some of our results strengthen and generalize those of a 2023 paper by Wang and Yang.

Hecke Relations for Eta Multipliers and Congruences of Higher-Order Smallest Parts Functions

Abstract

We derive identities from Hecke operators acting on a family of Eisenstein-eta quotients, yielding congruences for their coefficients modulo powers of primes. As an application we derive systematic congruences for several higher-order smallest parts functions modulo prime powers, resolving a question of Garvan for these cases. We also relate moments of cranks and ranks to the partition function modulo prime powers. Some of our results strengthen and generalize those of a 2023 paper by Wang and Yang.
Paper Structure (6 sections, 18 theorems, 77 equations)

This paper contains 6 sections, 18 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Hecke Relations for Eta Multipliers
  3. Consquences for Partition Statistics
  4. 4th and 6th Moments of Cranks
  5. 4th and 6th Moments of Ranks
  6. Proof of Theorem \ref{['thm: spt-main']}

Key Result

Theorem 1.1

Let $\delta_{j,\ell}$ be the Kronecker delta symbol. If $\ell\geq 5$ is prime and $\left(\frac{-n}{\ell} \right)=1$ we have

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 15 more