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A124175 Decimal expansion of Product_{primes p} ((p-1)/p)^(1/p)). 12

%I

%S 5,5,9,8,6,5,6,1,6,9,3,2,3,7,3,4,8,5,7,2,3,7,6,2,2,4,4,2,2,3,4,1,6,7,

%T 1,7,2,5,7,6,6,6,3,7,0,2,1,2,9,0,6,0,3,9,5,5,4,2,3,3,9,3,3,9,3,5,2,0,

%U 3,1,7,1,7,9,7,5,9,1,5,9,3,6,2,7,6,5,4,0,9,5,0,6,3,0,6,6,5,4,7

%N Decimal expansion of Product_{primes p} ((p-1)/p)^(1/p)).

%C This might be interpreted as the expected value of phi(n)/n for very large n.

%H Mathoverflow, <a href="http://mathoverflow.net/questions/230318/asymptotics-of-product-of-eulers-totient-function-a001088">Asymptotics of product of Euler's totient function</a>, 2016.

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

%F exp(-suminf(h=1, primezeta(h+1)/h)). - _Robert Gerbicz_

%F {Notation not clear. Is this perhaps exp(-Sum_{h=1..oo} primezeta(h+1)/h) ? - _N. J. A. Sloane_, Oct 08 2017)

%F Equals lim_{n->infinity} (A001088(n)/n!)^(1/n). - _Vaclav Kotesovec_, Feb 05 2016

%e 0.5598656169323734857237622442234167172576663702129060395542339339\

%e 352031717975915936276540950630665470795373094197373037280781542375...

%t digits = 100; s = Exp[-NSum[PrimeZetaP[h+1]/h, {h, 1, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> 3 digits]]; RealDigits[s, 10, digits][[1]] (* _Jean-Fran├žois Alcover_, Dec 07 2015, after _Robert Gerbicz_ *)

%o (PARI) default(realprecision,256);(f(k)=return(sum(n=1,512,moebius(n)/n*log(zeta(k*n)))));exp(sum(h=1,512,-1/h*f(h+1))) /* _Robert Gerbicz_ */

%o (PARI) exp(-suminf(m=2,log(zeta(m))*sumdiv(m,k,if(k<m,moebius(k)/(m-k),0)))) /* _Martin Fuller_ */

%Y Cf. A126226, A085548, A085541, A085964 - A085969, A242624, A272028.

%K nonn,cons

%O 0,1

%A _David W. Wilson_, Dec 05 2006

%E _Robert Gerbicz_ computed this to 130 decimal places.

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Last modified July 4 12:18 EDT 2020. Contains 335448 sequences. (Running on oeis4.)