Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms.

Eisenstein series for the modular group

[[Image:Gee_three_real.jpeg|thumb|The real part of G6 as a function of q on the unit disk. Negative numbers are black.]] [[Image:Gee_three_imag.jpeg|thumb|The imaginary part of G6 as a function of q on the unit disk.]] Let τ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series G2k(τ) of weight 2k , where k ≥ 2 is an integer, by the following series: This series absolutely converges to a holomorphic function of τ in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at . It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its SL(2, \mathbb{Z}) -covariance. Explicitly if a, b, c, d ∈ \mathbb{Z} and ad − bc = 1 then 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. For the series converges but it is not a modular form.

Relation to modular invariants

The modular invariants g2 and g3 of an elliptic curve are given by the first two Eisenstein series: The article on modular invariants provides expressions for these two functions in terms of theta functions.

Recurrence relation

Any holomorphic modular form for the modular group can be written as a polynomial in G4 and G6 . Specifically, the higher order G2k can be written in terms of G4 and G6 through a recurrence relation. Let , so for example, and . Then the dk satisfy the relation for all n ≥ 0 . Here, n \choose k is the binomial coefficient. The dk occur in the series expansion for the Weierstrass's elliptic functions:

Fourier series

[[Image:Eisenstein_4.jpg|right|thumb| G4 ]] [[Image:Eisenstein_6.jpg|right|thumb| G6 ]] [[Image:Eisenstein_8.jpg|right|thumb| G8 ]] [[Image:Eisenstein_10.jpg|right|thumb| G10 ]] [[Image:Eisenstein_12.jpg|right|thumb| G12 ]] [[Image:Eisenstein_14.jpg|right|thumb| G14 ]] Define . (Some older books define q to be the nome , but is now standard in number theory.) Then the Fourier series of the Eisenstein series is where the coefficients c2k are given by Here, Bn are the Bernoulli numbers, ζ(z) is Riemann's zeta function and σp(n) is the divisor sum function, the sum of the pth powers of the divisors of n. In particular, one has The summation over q can be resummed as a Lambert series; that is, one has for arbitrary complex and a. When working with the q-expansion of the Eisenstein series, this alternate notation is frequently introduced:

Identities involving Eisenstein series

As theta functions

Source: Given , let and define the Jacobi theta functions which normally uses the nome eπiτ , where θm and ϑij are alternative notations. Then we have the symmetric relations, Basic algebra immediately implies an expression related to the modular discriminant, The third symmetric relation, on the other hand, is a consequence of and a4 − b4 + c4 = 0 .

Products of Eisenstein series

Eisenstein series form the most explicit examples of modular forms for the full modular group SL(2, \mathbb{Z}) . Since the space of modular forms of weight 2k 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 q-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 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 [[E8 lattice|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.

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 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, 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 OK to be the ring of integers of a totally real algebraic number field K, one then defines the Hilbert–Blumenthal modular group as PSL(2,OK) . One can then associate an Eisenstein series to every cusp of the Hilbert–Blumenthal modular group.