Dirichlet Beta Function

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

			(1)
			(2)

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

(3)

The beta function can be defined over the whole complex plane using analytic continuation,

(4)

where is the gamma function.

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

The beta function can be evaluated directly special forms of arguments as

			(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)

(Guillera and Sondow 2005).

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.