The real part of G6 as a function of q on the unit disk. Negative numbers are black.
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 seriesG2k(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 τ = i∞. It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its SL(2, ℤ)-invariance. Explicitly if a, b, c, d ∈ ℤ and ad − bc = 1 then
If ad − bc = 1 then
is a bijection ℤ2 → ℤ2, i.e.:
Overall, if ad − bc = 1 then
and G2k is therefore a modular form of weight 2k. Note that it is important to assume that k ≥ 2, otherwise it would be illegitimate to change the order of summation, and the SL(2, ℤ)-invariance would not hold. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for k = 1, although it would only be a quasimodular form.
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 dk = (2k + 3)k! G2k + 4, so for example, d0 = 3G4 and d1 = 5G6. Then the dk satisfy the relation
Eisenstein series form the most explicit examples of modular forms for the full modular group SL(2, ℤ). Since the space of modular forms of weight 2k has dimension 1 for 2k = 4, 6, 8, 10, 14, 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:
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:
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
Srinivasa Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation. Let
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
^Ramanujan, Srinivasa (1962). "On certain arithmetical functions". Collected Papers. New York, NY: Chelsea. pp. 136–162.
^Melfi, Giuseppe (1998). "On some modular identities". Number Theory, Diophantine, Computational and Algebraic Aspects: Proceedings of the International Conference held in Eger, Hungary. Walter de Grutyer & Co. pp. 371–382.
Akhiezer, Naum Illyich (1970). "Elements of the Theory of Elliptic Functions" (in Russian). Moscow. Cite journal requires |journal= (help) Translated into English as Elements of the Theory of Elliptic Functions. AMS Translations of Mathematical Monographs 79. Providence, RI: American Mathematical Society. 1990. ISBN0-8218-4532-2.