Enumeration of the twin-prime pairs to 1e16 Thomas R. Nicely

Freeware copyright (c) 2010 Thomas R. Nicely. Released into the public domain by the author, who disclaims any legal liability arising from its use.

Last updated 1000 GMT 18 January 2010.

This is a table of the values of pi_2(x), the count of twin-prime pairs (q, q+2) such that q <= x. The first three twin-prime pairs are (3, 5), (5, 7), and (7, 11). Also provided are the values of the related functions delta_2(x), S_2(x), and F_2(x). The values in this table were obtained by the author's direct and explicit generation and enumeration of the primes.

The domain of this table consists of each decade from 10 through 1e12, then each 1e12 to 1e16. See also Enumeration of the twin-prime pairs from 1e16 to 2e16.

The most extensive counts of pi_2(x) of which I am aware have been compiled by Tomás Oliveira e Silva. However, Silva's compilations do not include the values of the related functions delta_2(x), S_2(x), and F_2(x).

Complete counts and reciprocal sums of the prime constellations from Nicely's computations (1993-2009), including the twin-prime pairs, are also available. These data files are very large (over 60MB each, even for the zipped versions), including more than two million data points from 0 to 2e16 at intervals of 1e10 or better.

Symbols are defined as follows; for further details, see the papers A new error analysis for Brun's constant and "Enumeration to $1.6 \times 10^{15}$ of the twin primes and Brun's constant."

• pi_2(x) = Number of twin-prime pairs (q, q+2) such that the smaller element q <= x. Note that, for example, pi_2(30) includes the pair (29, 31) in its count. Although perhaps counterintuitive, this is the convention dictated by prior usage in the literature.
• delta_2(x) = Li_2(x) - pi_2(x) . A positive value indicates a deficit of twin-prime pairs (compared to the theoretical estimate); a negative value indicates a surplus of twin-prime pairs. Referred to as r_3(x) in some papers.
• Li_2(x) = integral(2*c_2/(ln t)^2, t, 2, x), the Hardy-Littlewood integral approximation for pi_2(x). Although this is the traditional formula, note that a slightly more accurate (for small x) approximation is produced by the asymptotically equivalent formula Li_2*(x) = integral(2*c_2/((ln(t+6))^2), t, 2, x) .
• c_2 = Hardy-Littlewood constant for twins = 0.66016 18158 46869 57392 78121 10014 55577 84326 23.... The kth Hardy-Littlewood constant (k > 1) is defined as c_k = prod((p^(k-1))*(p-k)/(p-1)^k, p; p prime, p > k) . The value c_2 is also referred to as the twin-primes constant; however, some authors use 2*c_2 as the twin-primes constant.
• S_2(x) = Sum of the reciprocals of all the twin-prime pairs (q, q+2) such that q <= x. Note that, for example, S_2(30) includes the term 1/31. Also, if x >= 5, the term 1/5 is included twice in S_2(x). Referred to as B(x) in some papers.
• F_2(x) = First order extrapolation of Brun's constant B_2 (often referred to as simply B) from S_2(x). B_2 is the limit of the sum of the reciprocals as x approaches +infinity: B_2 \approx F_2(x) = S_2(x) + 4c_2/ln(x) . Referred to as B*(x) in some papers.
• The values given for pi_2(x) are believed to be exact. The values for delta_2(x), S_2(x), and F_2(x) are believed to be correct to all digits shown, except for a possible (rounding) error of one ulp (one unit in the last decimal place or least significant digit).
• Please inform me of any errors you discover in these tables.