Dirichlet Beta Function

The Dirichlet beta function, often shortened to Dirichlet beta, is defined for by the sum

			(1)
			(2)

where is the Lerch transcendent. The series converges absolutely for . The beta function can be written in terms of the Hurwitz zeta function by

(3)

At the difference of Hurwitz zeta functions is interpreted by its limit. The beta function extends to an entire function on the whole complex plane using analytic continuation,

(4)

where is the gamma function and removable singularities on the right are evaluated by limits.

The Dirichlet beta function is implemented in the Wolfram Language as DirichletBeta[x].

For a nonnegative integer , the beta function has the special values

			(5)
			(6)
			(7)

where is an Euler number.

Particular values for are

			(8)
			(9)
			(10)
			(11)

where is Catalan's constant and is the polygamma function. For , 3, 5, ..., , where the multiples are 1/4, 1/32, 5/1536, 61/184320, ... (OEIS A046976 and A053005).

It is involved in the integral

(12)

for (Guillera and Sondow 2008).

Rivoal and Zudilin (2003) proved that at least one of the seven numbers , , , , , , and is irrational.

The derivative can also be computed analytically at a number of integer values of including

			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)

(OEIS A378910, A113847, and A078127), where is Catalan's constant, is the gamma function, and is the Euler-Mascheroni constant.

A nice sum involving is given by

(20)

for a positive integer.