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 Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight , where is an integer, by the following series: This series absolutely converges to a holomorphic function of 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 -invariance. Explicitly if and then Relation to modular invariants The modular invariants and 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 and . Specifically, the higher order can be written in terms of and through a recurrence relation. Let , so for example, and . Then the satisfy the relation for all . Here, is the binomial coefficient. The occur in the series expansion for the Weierstrass's elliptic functions: Fourier series Define . (Some older books define to be the nome , but is now standard in number theory.) Then the Fourier series of the Eisenstein series is where the coefficients are given by Here, are the Bernoulli numbers, is Riemann's zeta function and is the divisor sum function, the sum of the th powers of the divisors of . In particular, one has The summation over can be resummed as a Lambert series; that is, one has for arbitrary complex and . When working with the -expansion of the Eisenstein series, this alternate notation is frequently introduced: Identities involving Eisenstein series As theta functions Given , let and define the Jacobi theta functions which normally uses the nome , where and 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 . Products of Eisenstein series Eisenstein series form the most explicit examples of modular forms for the full modular