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Gieseking's Constant

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Gieseking's constant is defined by

G=int_0^(2pi/3)ln(2cos(1/2x))dx
(1)
=Cl_2(1/3pi)
(2)
=(3sqrt(3))/4[1-sum_(k=0)^(infty)1/((3k+2)^2)+sum_(k=1)^(infty)1/((3k+1)^2)]
(3)
=-1/(36)i[pi^2-36Li_2(-(-1)^(2/3))]
(4)
=1/2i[Li_2((-1)^(-1/3))-Li_2((-1)^(1/3))]
(5)
=(9-psi_1(2/3)+psi_1(4/3))/(4sqrt(3))
(6)
=1.01494160640965...,
(7)

(OEIS A143298), where Cl_2(x) is Clausen's integral, Li_2(x) is a dilogarithm, and psi_1(x) is a trigamma function. The powers of -1 use their principal values.

T. Lupu (pers. comm., Jan. 12, 2026) supplied the rapidly convergent series

G=(3sqrt(3))/5-(2sqrt(3))/5sum_(n=1)^infty(nzeta(2n+1))/(36^n),
(8)

where zeta(s) is the Riemann zeta function.

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References

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 232-233, 2003.Sloane, N. J. A. Sequence A143298 in "The On-Line Encyclopedia of Integer Sequences."

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Gieseking's Constant

Cite this as:

Weisstein, Eric W. "Gieseking's Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GiesekingsConstant.html

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