Dirichlet beta function on arvau

Wikipedia / Dirichlet beta function

In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.

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:

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

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 are the primes of the form (5,13,17,...) and are the primes of the form (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

is the gamma function. It was conjectured by Euler in 1749 and proved by Malmsten in 1842.

, the following equation holds:

is the n-th Euler Number. This yields:

are irrational.

The even beta values may be given in terms of the polygamma functions and the Bernoulli numbers:

For every positive integer k:

For negative odd integers, the function is zero:

For every negative even integer it holds:

It further is:

We have:

being Euler's constant being the Lemniscate constant being Catalan's constant. The last identity was derived by Malmsten in 1842.

J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.