# Ramanujan’s partition generating functions modulo ℓ

## TL;DR

This paper provides a new proof of Ramanujan's partition congruences modulo 5, 7, and 11 using advanced number theory techniques.

## Contribution

The paper introduces a novel expression for partition generating functions modulo primes using Hecke traces of Dirichlet series.

## Key findings

- A new formula for partition generating functions modulo primes ℓ ≥ 5 is derived.
- The formula connects partition functions to Hecke traces of ℓ-ramified Dirichlet series.
- This provides a unified proof of Ramanujan's congruences modulo 5, 7, and 11.

## Abstract

For the partition function p(n), Ramanujan proved the striking identities \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \mathcal {P}_5(q):=\sum _{n\ge 0} p(5n+4)q^n&=5\prod _{n\ge 1} \frac{\left( q^5;q^5\right) _{\infty }^5}{(q;q)_{\infty }^6},\\ \mathcal {P}_{7}(q):=\sum _{n\ge 0} p(7n+5)q^n&=7\prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^3}{(q;q)_{\infty }^4}+49q \prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^7}{(q;q)_{\infty }^8}, \end{aligned} \end{aligned}$$\end{document}P5(q):=∑n≥0p(5n+4)qn=5∏n≥1q5;q5∞5(q;q)∞6,P7(q):=∑n≥0p(7n+5)qn=7∏n≥1q7;q7∞3(q;q)∞4+49q∏n≥1q7;q7∞7(q;q)∞8,where \documentclass[12pt]{minimal}
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\begin{document}$$(q;q)_{\infty }:=\prod _{n\ge 1}(1-q^n).$$\end{document}(q;q)∞:=∏n≥1(1-qn). As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes \documentclass[12pt]{minimal}
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\begin{document}$$\ell \ge 5,$$\end{document}ℓ≥5, closed form expressions of the power series \documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal {P}_{\ell }(q):=\sum _{n\ge 0} p(\ell n-\delta _{\ell })q^n\pmod {\ell }, $$\end{document}Pℓ(q):=∑n≥0p(ℓn-δℓ)qn(modℓ),where \documentclass[12pt]{minimal}
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\begin{document}$$\delta _{\ell }:=\frac{\ell ^2-1}{24}.$$\end{document}δℓ:=ℓ2-124. In this paper, we prove that \documentclass[12pt]{minimal}
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\begin{document}$$ \mathcal {P}_{\ell }(q)\equiv c_{\ell } \dfrac{\mathcal {T}_{\ell }(q)}{\left( q^\ell ; q^\ell \right) _\infty } \pmod {\ell }, $$\end{document}Pℓ(q)≡cℓTℓ(q)qℓ;qℓ∞(modℓ),where \documentclass[12pt]{minimal}
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\begin{document}$$c_{\ell }\in \mathbb Z$$\end{document}cℓ∈Z is explicit and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {T}}_{\ell }(q)$$\end{document}Tℓ(q) is the generating function for the Hecke traces of \documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}ℓ-ramified values of special Dirichlet series for weight \documentclass[12pt]{minimal}
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\begin{document}$$\ell -1$$\end{document}ℓ-1 cusp forms on \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{SL}_2(\mathbb Z)$$\end{document}SL2(Z). This is a new proof of Ramanujan’s congruences modulo 5, 7, and 11, as there are no nontrivial cusp forms of weight 4, 6, and 10.