Eisenstein series

Summary

The Eisenstein series $G_k$ of even weight $k \ge 4$ is the sum of $(m\tau + n)^{-k}$ over all pairs of integers $(m, n) \neq (0, 0)$. It is the simplest modular form of weight $k$ for the modular group that is not a cusp form. Its $q$-expansion involves the Riemann zeta function and the divisor sums $\sigma_{k-1}(n)$. The series $E_4$ and $E_6$ generate the ring of all modular forms for $\mathrm{SL}_2(\Z)$.