A344786 - OEIS

%I #5 May 28 2021 17:15:16

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

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

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

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

%C Deshouillers and Iwaniec (2008) proved that the sequence of geometric mean values of the Euler totient function, A001088(n)^(1/n) = (Product_{k=1..n} phi(k))^(1/n), is uniformly distributed modulo 1 if and only if this constant is irrational. They noted that Richard Bumby showed that if it is rational, then its denominator has at least 20 decimal digits.

%H Jean-Marc Deshouillers and Henryk Iwaniec, <a href="http://pcwww.liv.ac.uk/~karpenk/JournalUDT/vol03/no1/DesIwa08-1.pdf">On the distribution modulo one of the mean values of some arithmetical functions</a>, Uniform Distribution Theory, Vol. 3, No. 1 (2008), pp. 111-124.

%H Mehdi Hassani, <a href="http://www.inmabb.criba.edu.ar/revuma/pdf/v54n1/v54n1a06.pdf">Uniform distribution modulo one of some sequences concerning the Euler function</a>, Rev. Un. Mat. Argentina, Vol. 54, No. 1 (2013), pp. 55-68.

%F Equals exp(-1 - Sum_{k>=2} P(k)/(k-1)), where P(s) is the prime zeta function.

%e 0.20596305028818635387967542823249746648587805934205...

%t $MaxExtraPrecision = 1000; m = 100; RealDigits[N[Exp[-1 - Sum[PrimeZetaP[k]/(k - 1), {k, 2, 1000}]], m + 1], 10, m][[1]]

%Y Cf. A000010, A001088, A068985.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, May 28 2021