A396373 Decimal expansion of the determinant of the Laplacian on S^6, the 6-dimensional unit sphere, with the standard metric induced by the R^7 Euclidean norm.

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

Offset

1,2

References

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, p. 472.

Links

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

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.

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. 727.

Formula

Equals (1/5) * exp(455/432 - zeta'(-5)/30 - 2*zeta'(-3) - 149*zeta'(-1)/30).

Equals (1/5) * exp(303733/453600 + 751*(gamma + log(2*Pi))/1890 - 149*zeta'(2)/(60*Pi^2) + 3*zeta'(4)/(2*Pi^4) - zeta'(6)/(8*Pi^6)), where gamma is Euler's constant (A001620).

Equals (1/5) * exp(1381/2160 - 2*zeta'(-3) - zeta'(-5)/30) * A^(149/30), where A is the Glaisher-Kinkelin constant (A074962).

Example

1.290018366027266297051264330431495830757884049324567...

Mathematica

RealDigits[Exp[455/432 - Zeta'[-5]/30 - 2*Zeta'[-3] - 149*Zeta'[-1]/30] / 5, 10, 120][[1]]

Prog

(PARI) exp(455/432 - zeta'(-5)/30 - 2*zeta'(-3) - 149*zeta'(-1)/30) / 5

Crossrefs

Determinant of the Laplacian on S^n: A212002 (n=1), A396369 (n=2), A396370 (n=3), A396371 (n=4), A396372 (n=5), this constant (n=6), A396374 (n=7), A396375 (n=8), A396376 (n=9).

Cf. A001620, A061444, A073002, A074962, A084448, A259068, A259070, A261506.

Sequence in context: A260548 A019315 A127558 * A324330 A197294 A385063

Adjacent sequences: A396370 A396371 A396372 * A396374 A396375 A396376

Keyword

nonn,cons

Author

Amiram Eldar, May 24 2026

Status

approved