A344786 Decimal expansion of (1/e) * Product_{p prime} (1 - 1/p)^(1/p).

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

Offset: 0

Author

Amiram Eldar, May 28 2021

Keywords

• nonn
• cons

Comments

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.

Examples

			0.20596305028818635387967542823249746648587805934205...
		

Links

• Jean-Marc Deshouillers and Henryk Iwaniec, On the distribution modulo one of the mean values of some arithmetical functions, Uniform Distribution Theory, Vol. 3, No. 1 (2008), pp. 111-124.
• Mehdi Hassani, Uniform distribution modulo one of some sequences concerning the Euler function, Rev. Un. Mat. Argentina, Vol. 54, No. 1 (2013), pp. 55-68.

Crossrefs

Cf. A000010, A001088, A068985.

Programs

Mathematica

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

Formula

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