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Dirichlet Beta Function

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The Dirichlet beta function is defined by the sum

beta(x)=sum_(n=0)^(infty)(-1)^n(2n+1)^(-x)
(1)
=2^(-x)Phi(-1,x,1/2),
(2)

where Phi(z,s,a) is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function zeta(x,a) by

beta(x)=1/(4^x)[zeta(x,1/4)-zeta(x,3/4)].
(3)

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

beta(1-z)=(2/pi)^zsin(1/2piz)Gamma(z)beta(z),
(4)

where Gamma(z) 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

beta(-2k)=1/2E_(2k)
(5)
beta(-2k-1)=0
(6)
beta(2k+1)=((-1)^kE_(2k))/(2(2k)!)(1/2pi)^(2k+1),
(7)

where E_n is an Euler number.

Particular values for beta(n) are

beta(1)=1/4pi
(8)
beta(2)=K
(9)
beta(3)=1/(32)pi^3
(10)
beta(4)=1/(768)[psi_3(1/4)-8pi^4],
(11)

where K is Catalan's constant and psi_n(x) is the polygamma function. For n=1, 3, 5, ..., beta(n)=rpi^n, where the multiples are 1/4, 1/32, 5/1536, 61/184320, ... (OEIS A046976 and A053005).

It is involved in the integral

int_0^1int_0^1([-ln(xy)]^s)/(1+x^2y^2)dxdy=Gamma(s+2)beta(s+2)
(12)

(Guillera and Sondow 2005).

Rivoal and Zudilin (2003) proved that at least one of the seven numbers beta(2), beta(4), beta(6), beta(8), beta(10), beta(12), and beta(14) is irrational.

The derivative beta^'(x)|_(x=n) can also be computed analytically at a number of integer values of n including

beta^'(-1)=(2K)/pi
(13)
=0.583121808...
(14)
beta^'(0)=ln[(Gamma^2(1/4))/(2pisqrt(2))]
(15)
=0.391594392...
(16)
beta^'(1)=sum_(n=1)^(infty)((-1)^(n+1)ln(2n+1))/((2n+1))
(17)
=1/4pi{gamma+2ln2+3lnpi-4ln[Gamma(1/4)]}
(18)
=0.192901316...
(19)

(OEIS A133922, A113847, and A078127), where K is Catalan's constant, Gamma(z) is the gamma function, and gamma is the Euler-Mascheroni constant.

A nice sum involving beta^'(n) is given by

sum_(k=1)^inftyln[((4k+1)^(1/(4k+1)^n))/((4k-1)^(1/(4k-1)^n))]=-beta^'(n)
(20)

for n a positive integer.

See also

Catalan's Constant, Dirichlet Eta Function, Dirichlet Lambda Function, Hurwitz Zeta Function, Legendre's Chi-Function, Lerch Transcendent, Riemann Zeta Function, Sierpiński Constant, Zeta Function

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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.

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Dirichlet Beta Function

Cite this as:

Weisstein, Eric W. "Dirichlet Beta Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DirichletBetaFunction.html

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