The twin primes constant
where the primes only.
This can be written as
(5)
where prime zeta function .
Flajolet and Vardi (1996, p. 6) give products with accelerated convergence
with
(8)
where Möbius function and Riemann zeta
function . The values of A001037 ). The
two products have geometric convergence factors
(9)
(OEIS A005597 ). Le Lionnais (1983, p. 30) calls
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
Explore with Wolfram|Alpha
References Finch, S. R. "Hardy-Littlewood Constants." §2.1 in Mathematical
Constants. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical
Constants." Unpublished manuscript. 1996. https://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. Ribenboim,
P. The
New Book of Prime Number Records. Ribenboim, P. The
Little Book of Big Primes. Riesel,
H. Prime
Numbers and Computer Methods for Factorization, 2nd ed. Shanks, D. Solved
and Unsolved Problems in Number Theory, 4th ed. Sloane, N. J. A. Sequences A001037 /M0116
and A005597 /M4056 in "The On-Line Encyclopedia
of Integer Sequences." Veritasium. "We're 99.9% Sure This Pattern
Is True, but No One Can Prove It." Jun. 14, 2026. https://www.youtube.com/watch?v=8HBDE-msUjw . Wrench,
J. W. "Evaluation of Artin's Constant and the Twin Prime Constant."
Math. Comput. 15 , 396-398, 1961. Referenced on Wolfram|Alpha Twin Primes Constant
Cite this as:
Weisstein, Eric W. "Twin Primes Constant."
From MathWorld https://mathworld.wolfram.com/TwinPrimesConstant.html
Subject classifications
(sometimes also denoted 
) is defined by
![product_(p>2; p prime)[1-1/((p-1)^2)]](/images/equations/TwinPrimesConstant/Inline5.svg)
![exp{sum_(p>2; p prime)ln[(p(p-2))/((p-1)^2)]}](/images/equations/TwinPrimesConstant/Inline11.svg)
![exp{sum_(p>2; p prime)[ln(1-2/p)-2ln(1-1/p)]},](/images/equations/TwinPrimesConstant/Inline14.svg)
s
in sums and products are taken over primes only.
This can be written as
![Pi_2=exp{sum_(n=2)^infty(2-2^n)/n[P(n)-2^(-n)]},](/images/equations/TwinPrimesConstant/NumberedEquation1.svg)
is the prime zeta function.
![product_(n=2)^(infty)[zeta(n)(1-2^(-n))]^(-I_n)](/images/equations/TwinPrimesConstant/Inline19.svg)
![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),](/images/equations/TwinPrimesConstant/Inline22.svg)
is the Möbius function and 
is the Riemann zeta
function. The values of 
for 
, 2, ... are 2, 1, 2, 3, 6,
9, 18, 30, 56, 99, ... (OEIS A001037). The
two products have geometric convergence factors 
and 
, respectively (Flajolet and
Vardi 1996, p. 6).
was computed to 45 digits by Wrench (1961) and Gourdon and Sebah list 60 digits.
the Shah-Wilson constant, and 
the twin prime constant (Le Lionnais 1983, p. 37).
