A013668 Decimal expansion of zeta(10).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• cons
• nonn

Examples

			1.0009945751278180853371459589003190170060195315644775172577889946362914...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Cf. A013662, A013664, A013666, A013670.

Programs

Mathematica

RealDigits[Zeta[10], 10, 100][[1]] (* Vincenzo Librandi, Feb 15 2015 *)

PARI

zeta(10) \\ Michel Marcus, Feb 20 2015

Formula

Equals Pi^10/93555.

zeta(10) = 4/3*2^10/(2^10 - 1)*( Sum_{n even} n^2*p(n)/(n^2 - 1)^11 ), where p(n) = 3*n^10 + 55*n^8 + 198*n^6 + 198*n^4 + 55*n^2 + 3 is a row polynomial of A091043. - Peter Bala, Dec 05 2013

zeta(10) = Sum_{n >= 1} (A010052(n)/n^5) = Sum_{n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^5 ). - Mikael Aaltonen, Feb 20 2015

zeta(10) = Product_{k>=1} 1/(1 - 1/prime(k)^10). - Vaclav Kotesovec, May 02 2020

From Wolfdieter Lang, Sep 16 2020: (Start)

zeta(10) = (1/9!)*Integral_{0..infinity} x^9/(exp(x) - 1). See Abramowitz-Stegun, 23.2.7., for s=10, p. 807. The value of the integral is (128/33)*Pi^10 = (3.6324091...)*10^5.

zeta(10) = (4/1448685)*Integral_{0..infinity} x^9/(exp(x) + 1). See Abramowitz-Stegun, 23.2.8., for s=10, p. 807. The value of the integral is (511/132)*Pi^10 = (3.625314565...)*10^5. (End)