A243262 - OEIS

A243262

Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

1, 0, 3, 0, 9, 1, 6, 7, 5, 2, 1, 9, 7, 3, 9, 2, 1, 1, 4, 1, 9, 3, 3, 1, 3, 0, 9, 6, 4, 6, 6, 9, 4, 2, 2, 9, 0, 6, 3, 3, 1, 9, 4, 3, 0, 6, 4, 0, 3, 4, 8, 7, 0, 6, 0, 2, 2, 7, 2, 6, 1, 7, 4, 1, 1, 4, 5, 1, 6, 6, 0, 6, 6, 9, 7, 8, 2, 9, 0, 4, 0, 5, 2, 9, 2, 9, 3, 1, 3, 6, 2, 5, 5, 4, 8, 0, 8, 8, 5

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OFFSET

1,3

COMMENTS

Also known as the second Bendersky constant.

This is likely the same as the constant B considered in section 3 of the Choi and Srivastava link. - R. J. Mathar, Oct 03 2016 [Confirmed. See the Formula section. Choi and Srivastava's value 1.03092... is incorrect in its last decimal place. - Amiram Eldar, May 22 2026]

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, p. 53, eq. (69).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..2002 (corrected by Sean A. Irvine)

Junesang Choi and H. M. Srivastava, Certain classes of series involving the zeta function, J. Math. Annal. Applic., Vol. 231, No. 1 (1999), 91-117.

Junesang Choi and H. M. Srivastava, Certain classes of series associated with the Zeta function and multiple gamma functions, Journal of Mathematical Analysis and Applications, Vol. 118, No. 1-2 (2000), pp. 87-109.

K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.

Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.

Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant

FORMULA

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.

A(0) = sqrt(2*Pi) (A019727),

A(1) = A = Glaisher-Kinkelin constant (A074962),

A(2) = exp(-zeta'(-2)) = exp(zeta(3)/(4*Pi^2)).

Equals exp(-A240966). - Vaclav Kotesovec, Feb 22 2015

Equals exp(lim_{n->oo} Sum_{k=1..n} k^2*log(k) - (n^3/3 + n^2/2 + n/6)*log(n) + n^3/9 - n/12) (Choi and Srivastava, 1999). - Amiram Eldar, May 22 2026

EXAMPLE

1.03091675219739211419331309646694229...

MATHEMATICA

RealDigits[Exp[Zeta[3]/(4*Pi^2)], 10, 99] // First

(* Alternative: *)

RealDigits[Exp[N[(BernoulliB[2]/4)*(Zeta[3]/Zeta[2]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)

PROG

(PARI) exp(zeta(3)/(4*Pi^2)) \\ Felix Fröhlich, Jun 27 2019

CROSSREFS

Cf. A051675, A055462, A240966, A255269.