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In mathematics, Eisenstein series are particular modular forms with infinite series expansions that may be written down directly. They are initially defined here for the modular group.
Let <math>\tau<math> be a complex number with strictly positive imaginary part. Define the Eisenstein series
- <math>G_{2k}(\tau)<math>
for each integer <math>k >1<math>
by:
- <math>
G_{2k}(\tau) = \sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{2k}}
<math>
It is a remarkable fact that the Eisenstein series is a modular form. Explicitly
- <math>
G_{2k} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} G_{2k}(\tau)
<math>
such that
- <math> a,b,c,d \in \mathbb{Z}<math>
and satisfy
- <math> ad-bc=1<math>,
and therefore is a modular form of weight <math>2k<math>. Any holomorphic modular form for the modular group can be written as a polynomial in <math>G_4<math> and <math>G_6<math>. Define the nome <math>q=e^{i\pi\tau}<math>.
The Fourier series of the Eisenstein series is
- <math>
G_{2k}(\tau) = 2\zeta(2k) \left(1+c_{2k}\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{2n} \right)
<math>
where the Fourier coefficients <math>c_{2k}<math> are given by
- <math>
c_{2k} = \frac{(2\pi i)^{2k}}{(2k-1)! \zeta(2k)},
<math>
<math>\zeta(z)<math> is Riemann's zeta function and the sigma function <math>\sigma_p(n)<math> is the sum of the <math>p<math>th powers of the divisors of <math>n<math>.
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