A238734 Log of twice the twin prime constant, C_2, log(2*A005597).

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

Offset: 0

Author

John W. Nicholson, Mar 03 2014

Keywords

• nonn
• cons
• less

Comments

The value occurs as term in equation (15) in the Wolf paper. - Ralf Stephan, Mar 28 2014

Examples

			0.2778768820732319619323108667032534203602062941473682988245270533677164980...
		

Links

• Marek Wolf, Nearest-neighbor-spacing distribution of prime numbers and quantum chaos, Phys. Rev. E 89, 022922 (2014); arXiv preprint, arXiv:1212.3841 [math.NT], 2012-2014.

Crossrefs

Cf. A005597, A114907.

Programs

Mathematica

digits = 113;
s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 50];
C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 50}];
RealDigits[Log[2 C2]][[1]][[1 ;; digits]] (* Jean-François Alcover, Feb 16 2019 *)

PARI

default(realprecision,1000);
result={175/256*prod(k=2, 500, (zeta(k)*(1-1/2^k)*(1-1/3^k)*(1-1/5^k)*(1-1/7^k))^(-sumdiv(k, d, moebius(d)*2^(k/d))/k))};log(2*result)

PARI

log(2 * prodeulerrat(1-1/(p-1)^2, 1, 3)) \\ Amiram Eldar, Mar 16 2021

Formula

Equals log(2*A005597).