# Dirichlet Lambda Function

The Dirichlet Lambda function Lam(x) = 1 + 3^−x + 5^−x + ... = Sum{k=1, infinity}[(2k−1)^−x], x > 1. Lambda is defined for all values of x except x = 1 where it is infinite.

For example, Lam(2) = 1 + 1/9 + 1/25 + 1/49 + ... = Pi^2/8 = 1.23370,05501,36169,82735... .

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

Lam(x) = Dirichlet Lambda function:

The Dirichlet Lambda function of x > 1 is defined by the infinite series 1 + 1/3^x + 1/5^x + 1/7^x + ... . It is evaluated by first computing the Zeta function and then using the identity:

Lam(x) = Zeta(x) * (1−2^(−x))

which is good for all x except x = 1 where Lam(x) is infinite. Lam(0) = 0. Lam(2) = Pi^2 / 8. Lam(x) = 0 for all negative even integers.

See: Dirichlet Lambda Function -- From MathWorld
And: Wolfram Function Evaluation -- Zeta (Lam(x) = (1−2^(−x))*Zeta(x))

This page accessed times since April 7, 2005.
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Changes last made on Monday, 06-Aug-07 20:47:24 PDT