Ramanujan's partition generating functions modulo ℓ

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.