Ramanujan's congruence primes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910354838126592 |
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| author | Parnoff, Ellise Raghuram, A. |
| author_facet | Parnoff, Ellise Raghuram, A. |
| contents | 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 $χ.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03345 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ramanujan's congruence primes Parnoff, Ellise Raghuram, A. Number Theory 11F33, 11F67 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 $χ.$ |
| title | Ramanujan's congruence primes |
| topic | Number Theory 11F33, 11F67 |
| url | https://arxiv.org/abs/2403.03345 |