Dirichlet beta function

The Dirichlet beta function is defined as or, equivalently, In each case, it is assumed that Re(s) > 0.

Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane:[1] Another equivalent definition, in terms of the Lerch transcendent, is: which is once again valid for all complex values of s. The Dirichlet beta function can also be written in terms of the polylogarithm function: Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function but this formula is only valid at positive integer values of

It is also the simplest example of a series non-directly related to

which can also be factorized as an Euler product, thus leading to the idea of Dirichlet character defining the exact set of Dirichlet series having a factorization over the prime numbers.

At least for Re(s) ≥ 1: where p≡1 mod 4 are the primes of the form 4n+1 (5,13,17,...) and p≡3 mod 4 are the primes of the form 4n+3 (3,7,11,...).

This can be written compactly as The functional equation extends the beta function to the left side of the complex plane Re(s) ≤ 0.

It is given by where Γ(s) is the gamma function.

It was conjectured by Euler in 1749 and proved by Malmsten in 1842.

[2] For every odd positive integer

This yields: For the values of the Dirichlet beta function at even positive integers no elementary closed form is known, and no method has yet been found for determining the arithmetic nature of even beta values (similarly to the Riemann zeta function at odd integers greater than 3).

It has been proven that infinitely many numbers of the form

[5] The even beta values may be given in terms of the polygamma functions and the Bernoulli numbers:[6] We can also express the beta function for positive

in terms of the inverse tangent integral: For every positive integer k:[citation needed] where

For negative odd integers, the function is zero: For every negative even integer it holds:[3] It further is: We have:[3]