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%I M4056 #107 Apr 03 2023 10:36:09
%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 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) on primes p such that 2p+1 is also prime.
%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. - _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. - _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
%C Cohen (1998, p. 7) referred to this number as the "twin prime and Goldbach constant" and noted that, conjecturally, the number of twin prime pairs (p,p+2) with p <= X tends to 2*C_2*X/log(X)^2 as X tends to infinity. - _Artur Jasinski_, Feb 01 2021
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 11.
%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 84-93.
%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).
%H Harry J. Smith, <a href="/A005597/b005597.txt">Table of n, a(n) for n = 0..1001</a>
%H Folkmar Bornemann, <a href="http://www.ams.org/notices/200305/fea-smale.pdf">PRIMES Is in P: Breakthrough for "Everyman"</a>, Notices Amer. Math. Soc., Vol. 50, No. 5 (May 2003), p. 549.
%H Paul S. Bruckman, <a href="https://fq.math.ca/Scanned/39-4/advanced39-4.pdf">Problem H-576</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 39, No. 4 (2001), p. 379; <a href="https://www.fq.math.ca/Scanned/40-4/advanced40-4.pdf">General IZE</a>, Solution to Problem H-576 by the proposer, ibid., Vol. 40, No. 4 (2002), pp. 383-384.
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=TwinPrimeConstant">twin prime constant</a>.
%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High-precision computation of Hardy-Littlewood constants</a>, (1998).
%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Mathematical Constants, Errata and Addenda</a>, arXiv:2001.00578 [math.HO], 2020, Sec. 2.1.
%H Philippe Flajolet and Ilan Vardi, <a href="http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta Function Expansions of Classical Constants</a>, Unpublished manuscript. 1996.
%H Daniel A. Goldston, Timothy Ngotiaoco and Julian Ziegler Hunts, <a href="https://doi.org/10.7169/facm/1602">The tail of the singular series for the prime pair and Goldbach problems</a>, Functiones et Approximatio Commentarii Mathematici, Vol. 56, No. 1 (2017), pp. 117-141; <a href="http://arxiv.org/abs/1409.2151">arXiv preprint</a>, arXiv:1409.2151 [math.NT], 2014.
%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 [math.NT], 2009-2011, 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://www.plouffe.fr/simon/constants/twinprime.txt">The twin primes constant</a>.
%H Pascal Sebah, <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>.
%H John W. Wrench, Jr., <a href="http://dx.doi.org/10.1090/S0025-5718-1961-0124305-0">Evaluation of Artin's constant and the twin-prime constant</a>, Math. Comp., Vol. 15, No. 76 (1961), pp. 396-398.
%F Equals Product_{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. - _Jonathan Sondow_, Nov 18 2009
%F Equals Sum_{k>=1} mu(2*k-1)/phi(2*k-1)^2, where mu is the Möbius function (A008683) and phi is the Euler totient function (A000010) (Bruckman, 2001). - _Amiram Eldar_, Jan 14 2022
%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 *)
%t digits = 105; f[n_] := -2*(2^n-1)/(n+1); C2 = Exp[NSum[f[n]*(PrimeZetaP[n+1] - 1/2^(n+1)), {n, 1, Infinity}, NSumTerms -> 5 digits, WorkingPrecision -> 5 digits]]; RealDigits[C2, 10, digits][[1]] (* _Jean-François Alcover_, Apr 16 2016, updated Apr 24 2018 *)
%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))
%o (PARI) prodeulerrat(1-1/(p-1)^2, 1, 3) \\ _Amiram Eldar_, Mar 12 2021
%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), A167864, A000010, A008683.
%K cons,nonn,nice
%O 0,1
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Nov 08 2001
%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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