Dirichlet eta function
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The Dirichlet eta function can be defined as
- <math>\eta(s) = \left(1-2^{1-s}\right) \zeta(s)<math>
where ζ is Riemann's zeta function. However, it can also be used to define the zeta function. It has a Dirichlet series expression, valid for any complex number s with positive real part, given by
- <math>\eta(s) = \sum_{n=1}^{\infty}{(-1)^{n-1} \over n^s}.<math>
While this is convergent only for s with positive real part, it is Abel summable for any complex number, which serves to define the eta function as an entire function, and shows the zeta function is meromorphic with a single pole at s = 1.
Equivalently, we may begin by defining
- <math>\eta(s) = \frac{1}{\Gamma(s)}\int_0^\infty \frac{x^s}{\exp(x)+1}\frac{dx}{x}<math>
which is also defined in the region of positive real part. This gives the eta function as a Mellin transform.
Hardy gave a simple proof of the functional equation for the eta function, which is
- <math>\eta(-s) = 2\pi^{-s-1} s \sin\left({\pi s \over 2}\right) \Gamma(s)\eta(s+1).<math>
From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.
Borwein's method
Peter Borwein used approximations involving Chebyshev polynomials to produce a method for efficient evaluation of the eta function. If <math>d_k = n\sum_{i=0}^k \frac{(n+i-1)!4^i}{(n-i)!(2i)!}<math> then
- <math>\eta(s) = -\frac{1}{d_n} \sum_{k=0}^{n-1}\frac{(-1)^k(d_k-d_n)}{(k+1)^s}+\gamma_n(s),<math>
where the error term γn is bounded by
- <math>\gamma_n(s) \le \frac{3}{(3+\sqrt{8})^n} (1+2|t|)\exp(|t|\pi/2)<math>
where <math>t = \Im(s)<math>.
References
Borwein, P., An Efficient Algorithm for the Riemann Zeta Function, Constructive experimental and nonlinear analysis, CMS Conference Proc. 27 (2000), 29-34 or http://www.cecm.sfu.ca/~pborwein/