A396369 Decimal expansion of the determinant of the Laplacian on S^2, the 2-dimensional unit sphere, with the standard metric induced by the R^3 Euclidean norm.

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

Offset

1,1

References

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, p. 471.

Links

Table of n, a(n) for n=1..105.

Junesang Choi, Determinant of Laplacian on S^3, Math. Japonica, Vol. 40, No. 1 (1994), pp. 155-166. See Theorem 4.1, p. 162.

Junesang Choi and H. M. Srivastava, An application of the theory of the double Gamma function, Kyushu Journal of Mathematics, Vol. 53, No. 1 (1999), pp. 209-222. See p. 219, eq. (3.14).

Junesang Choi, Multiple Gamma Functions and Their Applications, in: G. Milovanović and M. Rassias (eds.), Analytic Number Theory, Approximation Theory, and Special Functions, Springer, New York, NY, 2014, pp. 93-129. See p. 124.

José Cunha and Pedro Freitas, Recurrence formulae for spectral determinants, Journal of Number Theory, Vol. 267 (2025), pp. 134-175; arXiv preprint, arXiv:2404.12114 [math.SP], 2024. See Corollary 2.9, p. 16.

William Duke and Özlem Imamoḡlu, Special values of multiple gamma functions, Journal de théorie des nombres de Bordeaux, Vol. 18, No. 1 (2006), pp. 113-123. See p. 116.

Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2024. See p. 22.

J. R. Quine and Junesang Choi, Zeta regularized products and functional determinants on spheres, The Rocky Mountain Journal of Mathematics, Vol. 26, No. 2 (1996), pp. 719-729; JSTOR link. See p. 726.

Formula

Equals exp(1/2 - 4*zeta'(-1)).

Equals exp(1/6 + (gamma + log(2*Pi))/3 - 2*zeta'(2)/Pi^2), where gamma is Euler's constant (A001620).

Equals exp(1/6) * A^4, where A is the Glaisher-Kinkelin constant (A074962).

Example

3.195311486059186083954018930320625869027025537218383...

Mathematica

RealDigits[Exp[1/6] * Glaisher^4, 10, 120][[1]]

Prog

(PARI) exp(1/2 - 4*zeta'(-1))

Crossrefs

Determinant of the Laplacian on S^n: A212002 (n=1), this constant (n=2), A396370 (n=3), A396371 (n=4), A396372 (n=5), A396373 (n=6), A396374 (n=7), A396375 (n=8), A396376 (n=9).

Cf. A001620, A061444, A073002, A074962, A084448.

Sequence in context: A270236 A391210 A140714 * A112932 A337680 A077895

Adjacent sequences: A396366 A396367 A396368 * A396370 A396371 A396372

Keyword

nonn,cons

Author

Amiram Eldar, May 24 2026

Status

approved