Dirichlet beta function


In mathematics, the Dirichlet 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.

Definition

The Dirichlet beta function is defined as
or, equivalently,
In each case, it is assumed that Re > 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.
Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function

Euler product formula

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 ≥ 1:
where are the primes of the form and are the primes of the form . This can be written compactly as

Functional equation

The functional equation extends the beta function to the left side of the complex plane Re ≤ 0. It is given by
where Γ is the gamma function.

Special values

Some special values include:
where G represents Catalan's constant, and
where in the above is an example of the polygamma function. More generally, for any positive integer k:
where represent the Euler numbers. For integer k ≥ 0, this extends to:
Hence, the function vanishes for all odd negative integral values of the argument.
For every positive integer k:
where is the Euler zigzag number.
Also it was derived by Malmsten in 1842 that
sapproximate value βOEIS
1/50.5737108471859466493572665
1/40.5907230564424947318659591
1/30.6178550888488520660725389
1/20.6676914571896091766586909
10.7853981633974483096156608
20.9159655941772190150546035
30.9689461462593693804836348
40.9889445517411053361084226
50.9961578280770880640063194
60.9986852222184381354416008
70.9995545078905399094963465-
80.9998499902468296563380671-
90.9999496841872200898213589-
100.9999831640261968774055407-

There are zeros at -1; -3; -5; -7 etc.