# Riemann Zeta Function

The Riemann Zeta function:

Zeta(x) = 1 + 2^−x + 3^−x + ... = Sum{k=1, infinity}[k^−x], x > 1.

Zeta is defined for all values of x except x = 1 where it is infinite.

For example, Zeta(2) = 1 + 1/4 + 1/9 + 1/16 + ... = Pi^2/6 = 1.64493,40668,48226,43647... .

Here are some notes from my program XPCalc - Extra Precision Floating-Point Calculator http://www.oocities.org/hjsmithh/download.html#XPCalc :

Zeta(x) = Riemann Zeta function:

The Riemann Zeta function of x > 1 is defined by the infinite series 1 + 1/2^x + 1/3^x + 1/4^x + ... .

It is evaluated by first computing the eta function and then using the identity:

Zeta(x) = Eta(x) * (2^x)/(2^x − 2)

which is good for all x except x = 1 where Zeta(x) is infinite. Zeta(0) = −1/2. Zeta(2) = Pi^2/6. Zeta(3) = Z = 1.202056903159594285399738161511449990765... . Zeta(x) = 0 for all negative even integers.

See: Riemann Zeta Function -- From MathWorld
And: Wolfram Function Evaluation -- Zeta

This page accessed times since April 6, 2005.
Page created by: hjsmithh@sbcglobal.net
Changes last made on Monday, 06-Aug-07 20:47:35 PDT