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Ramanujan's congruence primes

Ellise Parnoff, A. Raghuram

Abstract

Ramanujan showed that $τ(p) \equiv p^{11}+1 \pmod{691}$, 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 $f_χ\equiv E^\circ_{6, χ} \pmod{67}$ between a cusp form $f_χ$ and an Eisenstein series $E^\circ_{6, χ}$ of weight $6$ on $Γ_0(5)$ with nebentypus character $χ.$

Ramanujan's congruence primes

Abstract

Ramanujan showed that , where is the -th Fourier coefficient of the unique normalized cusp form of weight and full level, and the prime appears in the numerator of for the Riemann zeta function . Searching for such congruences, it is shown that the prime appears in the numerator of , where is the unique nontrivial quadratic Dirichlet character modulo and its Dirichlet -function, giving rise to a congruence between a cusp form and an Eisenstein series of weight on with nebentypus character
Paper Structure (12 sections, 5 theorems, 28 equations)

This paper contains 12 sections, 5 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Special values of the Riemann $\zeta$-function
  3. Special values of Dirichlet $L$-functions
  4. Computing $B_{2m, \chi}$
  5. Computing $B_{2m, \chi}$ via Bernoulli polynomials
  6. Computing $B_{2m, \chi}$ via recursion
  7. Values of $B_{2m, \chi}$ for $p=5$
  8. Ramanujan type congruence in $M_6(\Gamma_0(5), \chi)$ modulo $67$
  9. The Hecke--Eisenstein series $E^{\circ }_{6, \chi} \in M_6(\Gamma_0(5), \chi)$
  10. The cusp form $f_\chi \in S_6(\Gamma_0(5), \chi)$
  11. Congruence mod $67$
  12. Some comments and acknowledgements

Key Result

Theorem 3.1

Let $\chi$ denote the unique nontrivial quadratic Dirichlet character modulo an odd prime $p\equiv 1 \pmod{4}.$ For any positive integer $m$ we have:

Theorems & Definitions (7)

  • Theorem 3.1: Leopoldt (1958)
  • Theorem 3.2: Carlitz
  • Remark 3.3
  • Theorem 3.5
  • Remark 3.6
  • Theorem 4.4: Sturm
  • Theorem 4.5