Dirichlet Beta Function

Min Max

Min   Max       Re       Im

The Dirichlet beta function 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 A133922, 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.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807-808, 1972.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 384, 1987.

Comtet, L. Problem 37 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 89, 1974.

Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005 http://arxiv.org/abs/math.NT/0506319.

Rivoal, T. and Zudilin, W. "Diophantine Properties of Numbers Related to Catalan's Constant." Math. Ann. 326, 705-721, 2003. http://www.mi.uni-koeln.de/~wzudilin/beta.pdf.

Sloane, N. J. A. Sequences A046976, A053005, A078127, A113847, and A133922 in "The On-Line Encyclopedia of Integer Sequences."

Spanier, J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25-33, 1987.

Mathews, J. and Walker, R. L. Mathematical Methods of Physics, 2nd ed. Reading, MA: W. A. Benjamin/Addison-Wesley, p. 57, 1970.