cp's OEIS Frontend

A013665 Decimal expansion of zeta(7).

%I A013665 #95 Apr 29 2025 04:54:53
%S A013665 1,0,0,8,3,4,9,2,7,7,3,8,1,9,2,2,8,2,6,8,3,9,7,9,7,5,4,9,8,4,9,7,9,6,
%T A013665 7,5,9,5,9,9,8,6,3,5,6,0,5,6,5,2,3,8,7,0,6,4,1,7,2,8,3,1,3,6,5,7,1,6,
%U A013665 0,1,4,7,8,3,1,7,3,5,5,7,3,5,3,4,6,0,9,6,9,6,8,9,1,3,8,5,1,3,2
%N A013665 Decimal expansion of zeta(7).
%C A013665 From _Dimitris Valianatos_, Apr 29 2020: (Start)
%C A013665 Let p_n = Product_{k >= 1, 4*k-1 is prime} (((4*k - 1)^n + 1) / ((4*k - 1)^n - 1)).
%C A013665 Then (2^(n + 1) / (2^n - 1)) * Sum_{k >= 1} 1 / (4*k - 3)^n = ((p_n + 1) / p_n) * Sum_{k >= 1} 1 / k^n = ((p_n + 1) / p_n) * zeta(n), n >= 3 odd number.
%C A013665 For n = 7, p_7 = 1.00091744947834007403796003463414...
%C A013665 The product (256 / 127) * Sum_{k >= 1} 1 / (4*k - 3)^7 = 2.01577429320860871987548541116538... is equal to the product ((p_7 + 1) / p_7) * Sum_{k >= 1} 1 / k^7 = 1.9990833914636834116748... * zeta(7) = 2.01577429320860871987548541116538... (End)
%D A013665 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.
%D A013665 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
%D A013665 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.
%H A013665 Jakob Ablinger, <a href="https://arxiv.org/abs/1908.06631">Proving two conjectural series for zeta(7) and discovering more series for zeta(7)</a>, arXiv:1908.06631 [math.CO], 2019.
%H A013665 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A013665 J. Borwein and D. Bradley, <a href="http://arXiv.org/abs/math.CA/0505124">Empirically determined Apéry-like formulas for zeta(4n+3)</a>, arXiv:math/0505124 [math.CA], 2005.
%H A013665 Michael J. Dancs and Tian-Xiao He, <a href="http://dx.doi.org/10.1016/j.jnt.2005.09.005">An Euler-type formula for zeta(2k+1)</a>, Journal of Number Theory, Volume 118, Issue 2, June 2006, Pages 192-199.
%H A013665 Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/zeta7.txt">Zeta(7) to 50000 digits</a>.
%H A013665 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap100.html">Zeta(7) to 512 places:sum(1/n^7, n=1..infinity)</a>.
%F A013665 zeta(7) = Sum_{n >= 1} (A010052(n)/n^(7/2)) = Sum_{n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^(7/2) ). - _Mikael Aaltonen_, Feb 22 2015
%F A013665 zeta(7) = Product_{k>=1} 1/(1 - 1/prime(k)^7). - _Vaclav Kotesovec_, Apr 30 2020
%F A013665 From _Artur Jasinski_, Jun 27 2020: (Start)
%F A013665 zeta(7) = (-1/840)*Integral_{x=0..1} log(1-x^6)^7/x^7.
%F A013665 zeta(7) = (1/720)*Integral_{x=0..oo} x^6/(exp(x)-1).
%F A013665 zeta(7) = (4/2835)*Integral_{x=0..oo} x^6/(exp(x)+1).
%F A013665 zeta(7) = (1/(182880*Zeta(1/2)^7))*(-61*Pi^7*zeta(1/2)^7 + 2880* zeta'(1/2)^7 - 10080*zeta(1/2)*zeta'(1/2)^5*zeta''(1/2) + 10080* zeta(1/2)^2*zeta'(1/2)^3*zeta''(1/2)^2 - 2520*zeta(1/2)^3*zeta'(1/2)* zeta''(1/2)^3 + 3360*zeta(1/2)^2*zeta'(1/2)^4*zeta'''(1/2) - 5040 zeta(1/2)^3*zeta'(1/2)^2*zeta''(1/2)*zeta'''(1/2) + 840*zeta(1/2)^4* zeta''(1/2)^2*zeta'''(1/2) + 560*zeta(1/2)^4*zeta'(1/2)*zeta'''(1/2)^3  - 840*zeta(1/2)^3*zeta'(1/2)^3*zeta''''(1/2) + 840*zeta(1/2)^4*zeta'(1/2)* zeta''(1/2)*zeta''''(1/2) - 140*zeta(1/2)^5*zeta'''(1/2)*zeta''''(1/2) + 168*zeta(1/2)^4*zeta'(1/2)^2*zeta'''''(1/2) - 84*zeta(1/2)^5*zeta''(1/2)* zeta'''''(1/2) - 28*zeta(1/2)^5*zeta'(1/2)*zeta''''''(1/2) + 4* zeta(1/2)^6*zeta'''''''(1/2)). (End)
%F A013665 Equals 19*Pi^7/56700 - 2*Sum_{k>=1} 1/(k^7*(exp(2*Pi*k) - 1)) [Grosswald] (see Finch). - _Stefano Spezia_, Nov 01 2024
%F A013665 From _Peter Bala_, Apr 27 2025: (Start)
%F A013665 zeta(7) = 1/7! * Integral_{x >= 0} x^7 * exp(x)/(exp(x) - 1)^2 dx = 2^6/(2^6 - 1) * 1/7! * Integral_{x >= 0} x^7 * exp(x)/(exp(x) + 1)^2 dx.
%F A013665 zeta(7) = 1/8! * Integral_{x >= 0} x^8 * exp(x)*(exp(x) + 1) /(exp(x) - 1)^3 dx = 1/ (2*3*7*15*63) * Integral_{x >= 0} x^8 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)
%e A013665 1.0083492773819228268397975498497967595998635605652387064172831365716014...
%t A013665 RealDigits[Zeta[7],10,120][[1]] (* _Harvey P. Dale_, Oct 23 2012 *)
%o A013665 (PARI) zeta(7) \\ _Michel Marcus_, Apr 17 2016
%Y A013665 Cf. A013661, A002117, A013662, A013663, A013664, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
%Y A013665 Cf. A023874, A023875, A248884.
%K A013665 nonn,cons
%O A013665 1,4
%A A013665 _N. J. A. Sloane_