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Twin Primes Constant

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The twin primes constant Pi_2 (sometimes also denoted C_2) is defined by

Pi_2=product_(p>2; p prime)[1-1/((p-1)^2)]
(1)
=product_(p>2; p prime)(p(p-2))/((p-1)^2)
(2)
=exp{sum_(p>2; p prime)ln[(p(p-2))/((p-1)^2)]}
(3)
=exp{sum_(p>2; p prime)[ln(1-2/p)-2ln(1-1/p)]},
(4)

where the ps in sums and products are taken over primes only. This can be written as

Pi_2=exp{(2-2^n)/n[P(n)-2^(-n)]},
(5)

where P(n) is the prime zeta function.

Flajolet and Vardi (1996) give series with accelerated convergence

Pi_2=product_(n=2)^(infty)[zeta(n)(1-2^(-n))]^(-I_n)
(6)
=3/4(15)/(16)(35)/(36)product_(n=2)^(infty)[zeta(n)(1-2^(-n))(1-3^(-n))(1-5^(-n))(1-7^(-n))]^(-I_n),
(7)

with

I_n=1/nsum_(d|n)mu(d)2^(n/d),
(8)

where mu(x) is the Möbius function. The values of I_n for n=1, 2, ... are 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, ... (OEIS A001037). Equation (7) has convergence like ∼(11/2)^(-n).

Pi_2 was computed to 45 digits by Wrench (1961) and Gourdon and Sebah list 60 digits.

Pi_2=0.6601618158...
(9)

(OEIS A005597). Le Lionnais (1983, p. 30) calls Pi_2 the Shah-Wilson constant, and 2Pi_2 the twin prime constant (Le Lionnais 1983, p. 37).

See also

Artin's Constant, Barban's Constant, Brun's Constant, Feller-Tornier Constant, Goldbach Conjecture, Heath-Brown-Moroz Constant, Mertens Constant, Murata's Constant, Prime Products, Quadratic Class Number Constant, Sarnak's Constant, Taniguchi's Constant, Twin Primes

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References

Finch, S. R. "Hardy-Littlewood Constants." §2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 84-94, 2003.Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. http://algo.inria.fr/flajolet/Publications/landau.ps.Gourdon, X. and Sebah, P. "Some Constants from Number Theory." http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html.Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44, 1-70, 1923.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 202, 1989.Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, p. 147, 1991.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 61-66, 1994.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 30, 1993.Sloane, N. J. A. Sequences A001037/M0116 and A005597/M4056 in "The On-Line Encyclopedia of Integer Sequences."Wrench, J. W. "Evaluation of Artin's Constant and the Twin Prime Constant." Math. Comput. 15, 396-398, 1961.

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Twin Primes Constant

Cite this as:

Weisstein, Eric W. "Twin Primes Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TwinPrimesConstant.html

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