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Ramanujan's congruence primes

Ellise Parnoff and A. Raghuram

Vol. 18 (2025), No. 1, 141–150
Abstract

Ramanujan showed that τ(p) p11 + 1(mod691), 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)π12 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)(π65), where χ is the unique nontrivial quadratic Dirichlet character modulo 5 and L(s,χ) its Dirichlet L-function, giving rise to a congruence fχ E6,χ(mod67) between a cusp form fχ and an Eisenstein series E6,χ of weight 6 on Γ0(5) with nebentypus character χ.

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