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

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

Author

David W. Wilson, Dec 05 2006

Keywords

• nonn
• cons

Comments

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

Examples

			0.5598656169323734857237622442234167172576663702129060395542339339\
352031717975915936276540950630665470795373094197373037280781542375...
		

Links

• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 164.
• Mathoverflow, Asymptotics of product of Euler's totient function, 2016.
• Eric Weisstein's World of Mathematics, Prime Zeta Function

Crossrefs

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

Programs

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 *)

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(kMartin Fuller */

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

Extensions

Robert Gerbicz computed this to 130 decimal places.