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%I #20 Jun 20 2021 02:47:00
%S 8,3,2,4,2,9,0,6,5,6,6,1,9,4,5,2,7,8,0,3,0,8,0,5,9,4,3,5,3,1,4,6,5,5,
%T 7,5,0,4,5,4,4,5,3,1,8,0,7,7,4,1,7,0,5,3,2,4,0,8,9,3,9,9,1,2,9,6,0,3,
%U 4,7,0,7,1,3,9,4,8,1,1,4,2,4,2,1,9,1,6,2,7,2,2,5,0,4,6,3,8,1
%N Decimal expansion of Rosser's constant.
%C Named after the American logician and mathematician John Barkley Rosser, Sr. (1907-1989). - _Amiram Eldar_, Jun 20 2021
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood constants, p. 86.
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>, Illinois J. Math., Vol. 6, No. 1 (1962), pp. 64-94, eq. (2.14).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimesConstant.html">Twin Primes Constant</a>.
%F 4*C_2/exp(2*EulerGamma), where C_2 is the twin primes constant.
%F Equals lim_{x->inf} Product_{2 < p <= x} (1-2/p)*log(x)^2.
%e 0.832429065661945278030805943531465575045445318077417053240893991296...
%t digits = 98; s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 60]; C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n) )*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 60}];
%t RealDigits[4*C2/Exp[2*EulerGamma], 10, digits] // First
%o (PARI) 4 * exp(-2*Euler) * prodeulerrat(1-1/(p-1)^2, 1, 3) \\ _Amiram Eldar_, Mar 17 2021
%Y Cf. A005597, A091724, A246061.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, May 25 2016
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