Ramanujan's congruence primes

Abstract

Ramanujan showed that $\tau (p)\equiv p^{11}+1(\mathrm{mod}691)$, where $\tau (n)$ 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 $\zeta (12)/{\pi }^{12}$ for the Riemann zeta function $\zeta (s)$. Searching for such congruences, it is shown that the prime $67$ appears in the numerator of $L(6,\chi )/({\pi }^6\sqrt{5})$, where $\chi$ is the unique nontrivial quadratic Dirichlet character modulo $5$ and $L(s,\chi )$ its Dirichlet $L$-function, giving rise to a congruence $f_{\chi }\equiv E_{6,\chi }^{\circ }(\mathrm{mod}67)$ between a cusp form $f_{\chi }$ and an Eisenstein series $E_{6,\chi }^{\circ }$ of weight $6$ on ${\Gamma }_0(5)$ with nebentypus character $\chi .$

To appear in Involve - a journal of Mathematics, this article is based on an undergraduate research project at Fordham University