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 A013665 Decimal expansion of zeta(7). 16
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS From Dimitris Valianatos, Apr 29 2020: (Start) Let p_n = Product_{k >= 1, 4*k-1 is prime} (((4*k - 1)^n + 1) / ((4*k - 1)^n - 1)). 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. For n = 7, p_7 = 1.00091744947834007403796003463414... 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) 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 Jakob Ablinger, Proving two conjectural series for zeta(7) and discovering more series for zeta(7), arXiv:1908.06631 [math.CO], 2019. 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]. J. Borwein and D. Bradley, Empirically determined ApĂ©ry-like formulas for zeta(4n+3), arXiv:math/0505124 [math.CA], 2005. Michael J. Dancs, Tian-Xiao He, An Euler-type formula for zeta(2k+1), Journal of Number Theory, Volume 118, Issue 2, June 2006, Pages 192-199. Simon Plouffe, Plouffe's Inverter, Zeta(7) to 50000 digits Simon Plouffe, Zeta(7) to 512 places:sum(1/n^7, n=1..infinity) FORMULA 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 zeta(7) = Product_{k>=1} 1/(1 - 1/prime(k)^7). - Vaclav Kotesovec, Apr 30 2020 From Artur Jasinski, Jun 27 2020 (Start): zeta(7) = (-1/840)*Integral_{x=0..1} log(1-x^6)^7/x^7. zeta(7) = (1/720)*Integral_{x=0..infinity} x^6/(exp(x)-1). zeta(7) = (4/2835)*Integral_{x=0..infinity} x^6/(exp(x)+1). 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). EXAMPLE 1.0083492773819228268397975498497967595998635605652387064172831365716014... MATHEMATICA RealDigits[Zeta[7], 10, 120][[1]] (* Harvey P. Dale, Oct 23 2012 *) PROG (PARI) zeta(7) \\ Michel Marcus, Apr 17 2016 CROSSREFS Cf. A023874, A023875, A248884. Sequence in context: A014549 A334064 A021549 * A209059 A202779 A328498 Adjacent sequences:  A013662 A013663 A013664 * A013666 A013667 A013668 KEYWORD nonn,cons AUTHOR STATUS approved

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Last modified October 23 04:16 EDT 2020. Contains 337962 sequences. (Running on oeis4.)