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Decimal expansion of zeta(2)*zeta(3)/zeta(6).
50

%I #77 Aug 21 2024 14:29:39

%S 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,

%T 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,

%U 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

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

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

%C 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

%C From _Stanislav Sykora_, Nov 14 2014: (Start)

%C 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.

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

%C 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.

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

%C The asymptotic mean of A005361. - _Amiram Eldar_, Apr 13 2020

%D Joe Roberts, Lure of the Integers, Mathematical Association of America, 1992. See p. 74.

%H Stanislav Sykora, <a href="/A082695/b082695.txt">Table of n, a(n) for n = 1..2000</a>

%H Paul T. Bateman, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa2115.pdf">The distribution of values of the Euler function</a>, Acta Arithmetica 21:1 (1972), pp. 329-345.

%H Olivier Bordellès and Benoit Cloitre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Bordelles/bord14.html">An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions</a>, J. Int. Seq. 16 (2013) #13.6.3.

%H Robert E. Dressler, <a href="http://projecteuclid.org/euclid.pjm/1102976431">A density which counts multiplicity</a>, Pacific J. Math. 34 (1970), pp. 371-378.

%H Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1945-08.pdf">Some remarks on Euler's ϕ function and some related problems</a>, Bull. Amer. Math. Soc. 51 (1945), pp. 540-544.

%H J. von zur Gathen et al., <a href="https://doi.org/10.5802/jtnb.436">Average order in cyclic groups</a>, J. Theor. Nombres Bordeaux, 16 (2004), 107-123. Lists several other papers where this constant arises.

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2317020">Powerful numbers</a>, Amer. Math. Monthly, Vol. 77 (1970), 848-852.

%H D. Handelman, <a href="http://arxiv.org/abs/1309.7417">Invariants for critical dimension groups and permutation-Hermite equivalence</a>, arXiv preprint arXiv:1309.7417 [math.AC], 2013.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientSummatoryFunction.html">Totient Summatory Function</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerfulNumber.html">Powerful Number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's totient function</a>.

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

%F The sum of the reciprocals of the powerful numbers, A001694. - _T. D. Noe_, May 03 2006

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

%F 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

%e 1.94359643682075920505707036257476343718785850176780571602663568890 ...

%t First@RealDigits[ Zeta[2]*Zeta[3]/Zeta[6], 10, 100]

%t RealDigits[ 315 Zeta[3]/(2 Pi^4), 10, 111][[1]] (* _Robert G. Wilson v_, Aug 11 2014 *)

%o (PARI) zeta(3)*315/2/Pi^4

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

%K cons,nonn

%O 1,2

%A _Benoit Cloitre_, Apr 12 2003

%E New definition from _Eric W. Weisstein_, May 04 2006