Dirichlet Eta Function

\eta(s) = \left(1-2^{1-s}
ight) \zeta(s)

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

:

\eta(s) = \sum_{n=1}^{\infty}{(-1)^{n-1} \over n^s}.

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
:

\eta(s) = rac{1}{\Gamma(s)}\int_0^\infty rac{x^s}{\exp(x)+1}rac{dx}{x}

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

:

\eta(-s) = 2\pi^{-s-1} s \sin\left({\pi s \over 2}
ight) \Gamma(s)\eta(s+1).

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

:

d_k = n\sum_{i=0}^k rac{(n+i-1)!4^i}{(n-i)!(2i)!}

then

:

\eta(s) = -rac{1}{d_n} \sum_{k=0}^{n-1}rac{(-1)^k(d_k-d_n)}{(k+1)^s}+\gamma_n(s),

where the error term γ_n is bounded by

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME

	MORE SHOPPING...

	CATEGORIES ABOUT DIRICHLET ETA FUNCTION

	zeta and l-functions

Information About

Dirichlet Eta Function