The 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
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.
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(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.
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
||approximate value β(s)
There are zeros at -1; -3; -5; -7 etc.