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, B_n are the Bernoulli numbers, ζ(z) is Riemann's zeta function and σ_p(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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