login
A124175
Decimal expansion of Product_{primes p} ((p-1)/p)^(1/p).
14
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, 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, 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
OFFSET
0,1
COMMENTS
This might be interpreted as the expected value of phi(n)/n for very large n.
LINKS
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 164.
Eric Weisstein's World of Mathematics, Prime Zeta Function
FORMULA
exp(-suminf(h=1, primezeta(h+1)/h)). - Robert Gerbicz
[Notation not clear. Is this perhaps exp(-Sum_{h=1..oo} primezeta(h+1)/h) ? - N. J. A. Sloane, Oct 08 2017]
Equals exp(1) * lim_{n->infinity} (A001088(n)/n!)^(1/n). - Vaclav Kotesovec, Feb 05 2016
EXAMPLE
0.5598656169323734857237622442234167172576663702129060395542339339\
352031717975915936276540950630665470795373094197373037280781542375...
MATHEMATICA
digits = 100; s = Exp[-NSum[PrimeZetaP[h+1]/h, {h, 1, Infinity}, WorkingPrecision -> digits+5, NSumTerms -> 3 digits]]; RealDigits[s, 10, digits][[1]] (* Jean-François Alcover, Dec 07 2015, after Robert Gerbicz *)
PROG
(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 */
(PARI) exp(-suminf(m=2, log(zeta(m))*sumdiv(m, k, if(k<m, moebius(k)/(m-k), 0)))) /* Martin Fuller */
KEYWORD
nonn,cons
AUTHOR
David W. Wilson, Dec 05 2006
EXTENSIONS
Robert Gerbicz computed this to 130 decimal places.
STATUS
approved