A396376 Decimal expansion of the determinant of the Laplacian on S^9, the 9-dimensional unit sphere, with the standard metric induced by the R^10 Euclidean norm.

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

Offset

0,1

Links

Table of n, a(n) for n=0..106.

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.

Formula

Equals (Pi/4) * exp(16399*zeta(3)/(10080*Pi^2) - 2087*zeta(5)/(1920*Pi^4) + 31*zeta(7)/(128*Pi^6) - zeta(9)/(128*Pi^8)).

Example

0.946733422184790063668405067763528569380126646809395...

Mathematica

RealDigits[(Pi/4) * Exp[16399*Zeta[3]/(10080*Pi^2) - 2087*Zeta[5]/(1920*Pi^4) + 31*Zeta[7]/(128*Pi^6) - Zeta[9]/(128*Pi^8)], 10, 120][[1]]

Prog

(PARI) (Pi/4) * exp(16399*zeta(3)/(10080*Pi^2) - 2087*zeta(5)/(1920*Pi^4) + 31*zeta(7)/(128*Pi^6) - zeta(9)/(128*Pi^8))

Crossrefs

Determinant of the Laplacian on S^n: A212002 (n=1), A396369 (n=2), A396370 (n=3), A396371 (n=4), A396372 (n=5), A396373 (n=6), A396374 (n=7), A396375 (n=8), this constant (n=9).

Cf. A002117, A013663, A013665, A013667.

Sequence in context: A309610 A198989 A255013 * A088539 A245084 A199054

Adjacent sequences: A396373 A396374 A396375 * A396377 A396378 A396379

Keyword

nonn,cons

Author

Amiram Eldar, May 24 2026

Status

approved