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 A005597 Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2). (Formerly M4056) 14

%I M4056

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

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

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

%N Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).

%C C_2 = Product_{ p prime > 2} (p * (p-2) / (p-1)^2) is the 2-tuple case of the Hardy-Littlewood prime k-tuple constant (part of First H-L Conjecture): C_k = Product_{ p prime > k} (p^(k-1) * (p-k) / (p-1)^k).

%C Although C_2 is commonly called the twin prime constant, it is actually the prime 2-tuple constant (prime pair constant) which is relevant to prime pairs (p, p+2m), m >= 1.

%C The Hardy-Littlewood Asymptotic Conjecture for Pi_2m(n), the number of prime pairs (p, p+2m), m >= 1, with p <= n, claims that asymptotically Pi_2m(n) ~ C_2(2m) * Li_2(n), where Li_2(n) = Integral_{2, n} (dx/log^2(x)) and C_2(2m) = 2 * C_2 * Product_{p prime > 2, p | m} (p-1)/(p-2), which gives: C_2(2) = 2 * C_2 as the prime pair (p, p+2) constant, C_2(4) = 2 * C_2 as the prime pair (p, p+4) constant, C_2(6) = 2* (2/1) * C_2 as the prime pair (p, p+6) constant, C_2(8) = 2 * C_2 as the prime pair (p, p+8) constant, C_2(10) = 2 * (4/3) * C_2 as the prime pair (p, p+10) constant, C_2(12) = 2 * (2/1) * C_2 as the prime pair (p, p+12) constant, C_2(14) = 2 * (6/5) * C_2 as the prime pair (p, p+14) constant, C_2(16) = 2 * C_2 as the prime pair (p, p+16) constant, ... And, for i >= 1, C_2(2^i) = 2 * C_2 as the prime pair (p, p+2^i) constant.

%C C_2 also occurs as part of other Hardy-Littlewood conjectures related to prime pairs, e.g. the Hardy-Littlewood conjecture concerning the distribution of the Sophie Germain primes (A156874), which happens to be primes p s.t. (p, 2p+1) is a prime pair.

%C Another constant related to the twin primes is Viggo Brun's constant B (sometimes also called the twin primes Viggo Brun's constant B_2) A065421, where B_2 = Sum (1/p + 1/q) as (p,q) runs through the twin primes.

%C Reciprocal of the Selberg-Delange constant A167864. See A167864 for additional comments and references. [From _Jonathan Sondow_, Nov 18 2009]

%C C_2 = Product_{prime p>2} (p-2)p/(p-1)^2 is an analog for primes of Wallis' product 2/pi = Product_{n=1 to oo} (2n-1)(2n+1)/(2n)^2. [From _Jonathan Sondow_, Nov 18 2009]

%C One can compute a cubic variant, product_{primes >2} (1-1/(p-1)^3) = 0.855392... = (2/3) * 0.6601618...* 1.943596... by multiplying this constant with 2/3 and A082695. - R. J. Mathar, Apr 03 2011

%D R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 11.

%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 84-93

%D Philippe Flajolet and Ilan Vardi, Zeta function Expansions of Classical constants, Feb. 18, 1996

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D J. W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., 15 (1961), 396-398.

%H Harry J. Smith, <a href="/A005597/b005597.txt">Table of n, a(n) for n=0,...,1001</a>

%H C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=TwinPrimeConstant">twin prime constant</a>

%H H. Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High-precision calculation of Hardy-Littlewood constants</a>, 1998. - From _N. J. A. Sloane_, Jan 26 2013

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/erradd.pdf">Mathematical Constants, Errata and Addenda</a>, Sec. 2.1. [From _Jonathan Sondow_, Nov 18 2009]

%H Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/publist.html">Zeta function expansions of some classical constants</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/0903.2514">Hardy-Littlewood constants embedded into infinite products over all positive integers</a>, arXiv:0903.2514 constant T_1^(2).

%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]

%H G. Niklasch, <a href="http://www.gn-50uma.de/alula/essays/Moree/Moree-details.en.shtml#t05-twin">Twin primes constant</a>

%H _Simon Plouffe_, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap90.html">The twin primes constant</a>

%H _Simon Plouffe_, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/twinprime.txt">The twin primes constant</a>

%H Pascal Sebah (pascal_sebah(AT)ds-fr.com), <a href="http://numbers.computation.free.fr/Constants/Primes/twin.html">Numbers, constants and computation</a> (gives 5000 digits)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimesConstant.html">Twin Primes Constant</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimeConjecture.html">Twin Prime Conjecture</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeConstellation.html">Prime Constellation</a>

%F prod(k>=2, (zeta(k)*(1-1/2^k))^(-sum(d/k, mu(d)*2^(k/d))/k)) - _Benoit Cloitre_, Aug 06 2003

%F Equals 1/A167864. [From _Jonathan Sondow_, Nov 18 2009]

%e 0.6601618158468695739278121100145557784326233602847334133194484233354056423...

%t s[n_] := (1/n)*N[ Sum[ MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[ (Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, 160}]; RealDigits[C2][[1]][[1 ;; 105]] (* _Jean-François Alcover_, Oct 15 2012, after PARI *)

%o (PARI) ?\p1000 ? 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))

%Y Cf. A065645 (continued fraction), A065646 (denominators of convergents to twin prime constant), A065647 (numerators of convergents to twin prime constant), A062270, A062271, A114907, A065418 (C_3).

%K cons,nonn,nice,changed

%O 0,1

%A _N. J. A. Sloane_.

%E More terms from _Vladeta Jovovic_, Nov 08 2001

%E Fixed my PARI program, had -n _Harry J. Smith_, May 19 2009

%E Commented and edited by _Daniel Forgues_, Jul 28 2009, Aug 04 2009, Aug 12 2009

%E PARI code removed by _D. S. McNeil_, Dec 26 2010

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