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.

Read more about this topic:  Eisenstein Series

Famous quotes containing the word series:

    “Rosalynn said, “Jimmy, if we could only get Prime Minister Begin and President Sadat up here on this mountain for a few days, I believe they might consider how they could prevent another war between their countries.” That gave me the idea, and a few weeks later, I invited both men to join me for a series of private talks. In September 1978, they both came to Camp David.”
    —Jimmy Carter (James Earl Carter, Jr.)