Eisenstein series

Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms.

Note that k ≥ 2 is necessary such that the series converges absolutely, whereas k needs to be even otherwise the sum vanishes because the (-m, -n) and (m, n) terms cancel out.

The dk occur in the series expansion for the Weierstrass's elliptic functions: Define q = e2πiτ.

(Some older books define q to be the nome q = eπiτ, but q = e2πiτ is now standard in number theory.)

The theta function of an eight-dimensional even unimodular lattice Γ is a modular form of weight 4 for the full modular group, which gives the following identities: for the number rΓ(n) of vectors of the squared length 2n in the root lattice of the type E8.

Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer n' as a sum of two, four, or eight squares in terms of the divisors of n. Using the above recurrence relation, all higher E2k can be expressed as polynomials in E4 and E6.

For example: Many relationships between products of Eisenstein series can be written in an elegant way using Hankel determinants, e.g. Garvan's identity where is the modular discriminant.

[8] Srinivasa Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation.

Following Ramanujan, to put these identities in the simplest form it is necessary to extend the domain of σp(n) to include zero, by setting Then, for example Other identities of this type, but not directly related to the preceding relations between L, M and N functions, have been proved by Ramanujan and Giuseppe Melfi,[10][11] as for example Automorphic forms generalize the idea of modular forms for general Lie groups; and Eisenstein series generalize in a similar fashion.