A082695 Decimal expansion of zeta(2)*zeta(3)/zeta(6).

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

Offset: 1

Author

Benoit Cloitre, Apr 12 2003

Keywords

• cons
• nonn

Comments

Equals the Dirichlet zeta-function Sum_{n>=1} A001615(n)/n^s at s=3. - R. J. Mathar, Apr 02 2011

Dressler shows that this is the average value of A014197, that is, the number of values m such that phi(m) <= n is asymptotically n times this constant. Erdős had shown earlier that this limit exists. - Charles R Greathouse IV, Nov 26 2013

From Stanislav Sykora, Nov 14 2014: (Start)

Equals lim_{n->infinity} (Sum_{k=1..n} k/phi(k))/n, i.e., the limit mean value of k/phi(k), where phi(k) is Euler's totient function.

Also equals lim_{n->infinity} (Sum_{k=1..n} 1/phi(k))/log(n).

Proofs are trivial using the formulas for Sum_{k=1..n} k/phi(k) and Sum_{k=1..n} 1/phi(k) listed in the Wikipedia link.

For the limit mean value of phi(k)/k, see A059956. (End)

The asymptotic mean of A005361. - Amiram Eldar, Apr 13 2020

Examples

			1.94359643682075920505707036257476343718785850176780571602663568890 ...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.7, p. 116.
• Joe Roberts, Lure of the Integers, Mathematical Association of America, 1992. See p. 74.

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• Paul T. Bateman, The distribution of values of the Euler function, Acta Arithmetica 21:1 (1972), pp. 329-345.
• Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3.
• Robert E. Dressler, A density which counts multiplicity, Pacific J. Math. 34 (1970), pp. 371-378.
• Paul Erdős, Some remarks on Euler's ϕ function and some related problems, Bull. Amer. Math. Soc. 51 (1945), pp. 540-544.
• J. von zur Gathen et al., Average order in cyclic groups, J. Theor. Nombres Bordeaux, 16 (2004), 107-123. Lists several other papers where this constant arises.
• S. W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
• D. Handelman, Invariants for critical dimension groups and permutation-Hermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013.
• Eric Weisstein's World of Mathematics, Totient Summatory Function.
• Eric Weisstein's World of Mathematics, Powerful Number.
• Wikipedia, Euler's totient function.

Crossrefs

Cf. A000010, A001615, A001694, A002117, A005117, A005361, A013661, A013664, A014197, A059956, A068468, A070243, A082696 (continued fraction), A157292.

Programs

Mathematica

First@RealDigits[ Zeta[2]*Zeta[3]/Zeta[6], 10, 100]
RealDigits[ 315 Zeta[3]/(2 Pi^4), 10, 111][[1]] (* Robert G. Wilson v, Aug 11 2014 *)

PARI

zeta(3)*315/2/Pi^4

Formula

Decimal expansion of Product_{p prime} (1+1/p/(p-1)) = zeta(2)*zeta(3)/zeta(6) = 1.94359643682075920505707...

The sum of the reciprocals of the powerful numbers, A001694. - T. D. Noe, May 03 2006

Equals A013661 * A002117 / A013664 = 1 / A068468 = zeta(3) * 315/(2*Pi^4) = zeta(3) * A157292.

Equals Sum_{k>=1} mu(k)^2/(k*phi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Euler totient function values, A000010). - Amiram Eldar, Aug 18 2020

Extensions

New definition from Eric W. Weisstein, May 04 2006