A396371 - OEIS

%I #5 May 24 2026 11:23:34

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

%T 2,8,8,7,2,7,2,5,7,5,6,2,7,8,3,3,5,7,3,6,5,4,0,4,0,7,4,8,6,4,4,4,7,4,

%U 3,8,5,0,8,6,5,2,8,9,5,7,8,9,8,5,2,6,6,2,9,2,9,4,2,9,5,7,9,5,6,2,5,4,7,3,8

%N Decimal expansion of the determinant of the Laplacian on S^4, the 4-dimensional unit sphere, with the standard metric induced by the R^5 Euclidean norm.

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

%H José Cunha and Pedro Freitas, <a href="https://doi.org/10.1016/j.jnt.2024.08.004">Recurrence formulae for spectral determinants</a>, Journal of Number Theory, Vol. 267 (2025), pp. 134-175; <a href="https://arxiv.org/abs/2404.12114">arXiv preprint</a>, arXiv:2404.12114 [math.SP], 2024. See Corollary 2.9, p. 16.

%H J. R. Quine and Junesang Choi, <a href="https://doi.org/10.1216/rmjm/118107208">Zeta regularized products and functional determinants on spheres</a>, The Rocky Mountain Journal of Mathematics, Vol. 26, No. 2 (1996), pp. 719-729; <a href="https://www.jstor.org/stable/44238421">JSTOR link</a>. See p. 726.

%F Equals (1/3) * exp(15/16 - 2*zeta'(-3)/3 - 13*zeta'(-1)/3).

%F Equals (1/3) * exp(1267/2160 + (16/45)*(gamma + log(2*Pi)) - 13*zeta'(2)/(6*Pi^2) + zeta'(4)/(2*Pi^4)), where gamma is Euler's constant (A001620).

%F Equals (1/3) * exp(83/144 - 2*zeta'(-3)/3) * A^(13/3), where A is the Glaisher-Kinkelin constant (A074962).

%e 1.736943483345117045218706484107328288727257562783357...

%t RealDigits[Exp[15/16 - 2*Zeta'[-3]/3 - 13*Zeta'[-1]/3] / 3, 10, 120][[1]]

%o (PARI) exp(15/16 - 2*zeta'(-3)/3 - 13*zeta'(-1)/3) / 3

%Y Determinant of the Laplacian on S^n: A212002 (n=1), A396369 (n=2), A396370 (n=3), this constant (n=4), A396372 (n=5), A396373 (n=6), A396374 (n=7), A396375 (n=8), A396376 (n=9).

%Y Cf. A001620, A061444, A073002, A074962, A084448, A259068, A261506.

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, May 24 2026