Ramanujan’s congruence primes

Parnoff, Ellise; Raghuram, A.

doi:10.2140/involve.2025.18.141

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Ramanujan's congruence primes

Ellise Parnoff and A. Raghuram

Vol. 18 (2025), No. 1, 141–150

DOI: 10.2140/involve.2025.18.141

Abstract

Ramanujan showed that $τ (p) \equiv p^{11} + 1 (\mod 691)$ , where $τ (n)$ is the $n$ -th Fourier coefficient of the unique normalized cusp form of weight $12$ and full level, and the prime $691$ appears in the numerator of $ζ (12) ∕ π^{12}$ for the Riemann zeta function $ζ (s)$ . Searching for such congruences, it is shown that the prime $67$ appears in the numerator of $L (6, χ) ∕ (π^{6} \sqrt{5})$ , where $χ$ is the unique nontrivial quadratic Dirichlet character modulo $5$ and $L (s, χ)$ its Dirichlet $L$ -function, giving rise to a congruence $f_{χ} \equiv E_{6, χ}^{\circ} (\mod 67)$ between a cusp form $f_{χ}$ and an Eisenstein series $E_{6, χ}^{\circ}$ of weight $6$ on $Γ_{0} (5)$ with nebentypus character $χ$ .

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Keywords

congruences, Dirichlet characters, $L$-functions

Mathematical Subject Classification

Primary: 11F33

Secondary: 11F67

Milestones

Received: 31 May 2023

Revised: 10 August 2023

Accepted: 12 August 2023

Published: 24 January 2025

Communicated by Ken Ono

Authors

Ellise Parnoff
Department of Mathematics Fordham University at Lincoln Center New York, NY United States

A. Raghuram
Department of Mathematics Fordham University at Lincoln Center New York, NY United States

Vol. 18, No. 1, 2025