group . Since the space of modular forms of weight has dimension 1 for , different products of Eisenstein series having those weights have to be equal up to a scalar multiple. In fact, we obtain the identities: Using the -expansions of the Eisenstein series given above, they may be restated as identities involving the sums of powers of divisors: hence and similarly for the others. The theta function of an eight-dimensional even unimodular lattice is a modular form of weight 4 for the full modular group, which gives the following identities: for the number of vectors of the squared length in the root lattice of the type . Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer ' as a sum of two, four, or eight squares in terms of the divisors of . Using the above recurrence relation, all higher can be expressed as polynomials in and . 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. Ramanujan identities Srinivasa Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation. Let then These identities, like the identities between the series, yield arithmetical convolution identities involving the sum-of-divisor function. Following Ramanujan, to put these identities in the simplest form it is necessary to extend the domain of to include zero, by setting Then, for example Other identities of this type, but not directly related to the preceding relations between , and functions, have been proved by Ramanujan and Giuseppe Melfi, as for example Generalizations Automorphic forms generalize the idea of modular forms for general Lie groups; and Eisenstein series generalize in a similar fashion. Defining to be the ring of integers of a totally real algebraic number field , one then defines the Hilbert–Blumenthal modular group as . One can then associate an Eisenstein series to every cusp of the Hilbert–Blumenthal modular group. References Further reading Translated into English as Mathematical series Modular forms Analytic number theory Fractals