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Ramanujan's partition generating functions modulo ℓ.

Kathrin Bringmann1, William Craig2, Ken Ono3

  • 1Department of Mathematics and Computer Science, Division of Mathematics, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany.

The Ramanujan Journal
|November 7, 2025
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Summary
This summary is machine-generated.

This study extends Ramanujan's partition identities to primes modulo ℓ. It proves a new congruence for the partition function p(n) modulo ℓ, connecting it to Hecke traces of cusp forms.

Keywords:
Modular formsPartition functionRamanujan’s partition congruences

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Area of Science:

  • Number Theory
  • Combinatorics
  • Modular Forms

Background:

  • Ramanujan's partition identities provide congruences for the partition function p(n) modulo 5 and 7.
  • These identities are crucial for understanding the behavior of p(n) for specific arithmetic progressions.

Purpose of the Study:

  • To generalize Ramanujan's findings by seeking closed-form expressions for the power series Pℓ(q) modulo ℓ for primes ℓ ≥ 5.
  • To establish a new congruence relation for the partition function modulo ℓ.

Main Methods:

  • The study utilizes the generating function for the partition function, p(n).
  • It involves analyzing power series Pℓ(q) and their behavior modulo ℓ.
  • The methods connect these series to Hecke traces of cusp forms.

Main Results:

  • A new congruence is proven: Pℓ(q) ≡ cℓ * Tℓ(q) / (qℓ; qℓ)∞ (mod ℓ).
  • Here, cℓ is an explicit integer, and Tℓ(q) represents Hecke traces of special Dirichlet series.
  • This result offers a new proof for Ramanujan's congruences modulo 5, 7, and 11.

Conclusions:

  • The derived congruence provides a unified framework for Ramanujan's partition congruences.
  • The connection to Hecke traces opens new avenues for research in number theory and modular forms.
  • The findings are significant for understanding the arithmetic properties of the partition function.