

A013663


Decimal expansion of zeta(5).


43



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

1,3


COMMENTS

In a widely distributed May 2011 email, Wadim Zudilin gave a rebuttal to v1 of Kim's 2011 preprint: "The mistake (unfixable) is on p. 6, line after eq. (3.3). 'Without loss of generality' can be shown to work only for a finite set of n_k's; as the n_k are sufficiently large (and N is fixed), the inequality for epsilon is false." In a May 2013 email, Zudilin extended his rebuttal to cover v2, concluding that Kim's argument "implies that at least one of zeta(2), zeta(3), zeta(4) and zeta(5) is irrational, which is trivial."  Jonathan Sondow, May 06 2013


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

Table of n, a(n) for n=1..99.
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].
Michael J. Dancs, TianXiao He, An Eulertype formula for zeta(2k+1), Journal of Number Theory, Volume 118, Issue 2, June 2006, Pages 192199.
Robert J. Harley, Zeta(3), Zeta(5), .., Zeta(99) 10000 digits (txt, 400 KB)
YongCheol Kim, zeta(5) is irrational, arXiv:1105.0730 [math.CA], 2011. [Jonathan Vos Post, May 4, 2011].
Simon Plouffe, Computation of Zeta(5)
Simon Plouffe, Zeta(5), the sum(1/n**5, n=1..infinity) to 512 digits
Simon Plouffe, Other interesting computations
Wikipedia, Zeta constant
W. Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational, Russ. Math. Surv., 56 (2001), 774776.


FORMULA

From Peter Bala, Dec 04 2013: (Start)
Definition: zeta(5) = sum {n >= 1} 1/n^5.
zeta(5) = 2^5/(2^5  1)*( sum {n even} n^5*p(n)*p(1/n)/(n^2  1)^6 ), where p(n) = n^2 + 3. See A013667, A013671 and A013675. (End)
zeta(5) = Sum_{n >= 1} (A010052(n)/n^(5/2)) = Sum_{n >= 1} ( (floor(sqrt(n))floor(sqrt(n1)))/n^(5/2) ).  Mikael Aaltonen, Feb 22 2015


EXAMPLE

1 + 1/32 + 1/243 + 1/1024 + 1/3125 + 1/7776 + 1/16807 + ... = 1.036927755143369926331365486457...


MATHEMATICA

RealDigits[Zeta[5], 10, 100][[1]] (* Alonso del Arte, Jan 13 2012 *)


PROG

(PARI) zeta(5) \\ Michel Marcus, Apr 17 2016


CROSSREFS

Cf. A002117, A013667, A013669, A013671, A013675, A013677, A243264, A255323.
Cf. A023872, A023873, A248882, A255050, A255052, A057528, A260404.
Sequence in context: A258329 A094561 A099679 * A245223 A180593 A271742
Adjacent sequences: A013660 A013661 A013662 * A013664 A013665 A013666


KEYWORD

nonn,cons


AUTHOR

N. J. A. Sloane


STATUS

approved



