A396370 - OEIS

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

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

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

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

%N Decimal expansion of the determinant of the Laplacian on S^3, the 3-dimensional unit sphere, with the standard metric induced by the R^4 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 Junesang Choi, <a href="https://www.researchgate.net/publication/266344792">Determinant of Laplacian on S^3</a>, Math. Japonica, Vol. 40, No. 1 (1994), pp. 155-166. See Theorem 4.1, p. 162.

%H Junesang Choi and H. M. Srivastava, <a href="https://doi.org/10.2206/kyushujm.53.209">An application of the theory of the double Gamma function</a>, Kyushu Journal of Mathematics, Vol. 53, No. 1 (1999), pp. 209-222. See p. 220, eq. (3.22).

%H Junesang Choi, <a href="https://doi.org/10.1007/978-1-4939-0258-3_4">Multiple Gamma Functions and Their Applications</a>, 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.

%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 William Duke and Özlem Imamoḡlu, <a href="https://doi.org/10.5802/jtnb.536">Special values of multiple gamma functions</a>, Journal de théorie des nombres de Bordeaux, Vol. 18, No. 1 (2006), pp. 113-123. See p. 116.

%H Steven Finch, <a href="https://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2024. See p. 22.

%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 Pi * exp(zeta(3)/(2*Pi^2)).

%F Equals (1/2) * exp(log(2*Pi) + zeta(3)/(2*Pi^2)).

%F Equals (1/2) * exp(-2*zeta'(-2) - 2*zeta'(0)).

%e 3.338851214151637978641073442361581066827638921418583...

%t RealDigits[Pi * Exp[Zeta[3]/(2*Pi^2)], 10, 120][[1]]

%o (PARI) Pi * exp(zeta(3)/(2*Pi^2))

%Y Determinant of the Laplacian on S^n: A212002 (n=1), A396369 (n=2), this constant (n=3), A396371 (n=4), A396372 (n=5), A396373 (n=6), A396374 (n=7), A396375 (n=8), A396376 (n=9).

%Y Cf. A002117, A075700, A240966.

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, May 24 2026