Eisenstein Series - Fourier Series

Fourier Series

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

G_{2k}(\tau) = 2\zeta(2k) \left(1+c_{2k}\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)

where the Fourier coefficients are given by

c_{2k} = \frac{(2\pi i)^{2k}}{(2k-1)! \zeta(2k)} = \frac {-4k}{B_{2k}} = \frac {2}{\zeta(1-2k)} .

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

and

Note the summation over q can be resummed as a Lambert series; that is, one has

for arbitrary complex |q| ≤ 1 and a. When working with the q-expansion of the Eisenstein series, the alternate notation

is frequently introduced.

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