Compare Viggo Brun's constant (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + (1/29 + 1/31) + ... (see A065421, A005597).
We can show that a(9)=6, a(10)=5, and a(11) is in the set {7, 8, 9}.
Proof: s1 = 0.237251776576249072... is the sum up to prime(499,000,000) and s2 = 0.237251776576250009... is the sum up to prime(500,000,000).
By using the fact that number of twin primes between the first 10^6*n primes and the first 10^6*(n+1) primes is decreasing (up to the first 2*10^9 primes), we conclude that the sum up to prime(2,000,000,000) is less than s2 + 1500*(s2-s1).
But since s2-s1 < 10^(-15), the sum up to prime(2*10^9) is less than s2 + 1.5*10^(-12) = 0.237251776576250009... + 1.5*10^(-12) = 0.237251776577550009... .
Hence the constant c is less than
0.237251776577550009... + lim(sum(1/k^2,{k, prime(2,000,000,001), n}, n -> infinity)
< 0.237251776577550009... + 2.12514*10^(-11)
< 0.237251776598801409.
So we have 0.237251776576250009 < c < 0.237251776598801409, hence a(9)=6, a(10)=5, and a(11) is in the set {7, 8, 9}.
I guess that a(11)=7. (End)
Given that the Hardy-Littlewood approximation to the number of twin prime pairs < y is
2 * C_2 * Integral_{x=2..y} dx/log(x)^2
where C_2 = 0.660161815846869573927812110014555778432623 (see A152051), we can estimate the size of the tail of the summation Sum(1/A001359(j)^2) + 1/A006512(j)^2) for twin primes > y as
t(y) = 2 * C_2 * Integral_{x>y} 2*dx/(x*log(x))^2.
Let s(y) be the sum of the squares of the reciprocals of all the twin primes <= y, and let s'(y) = s(y) + t(y) be the result of adding to the actual value s(y) the estimated tail size t(y). Evaluating s(y), t(y), and s'(y) at y = 2^d for d = 20..33 gives
.
d s(2^d) t(2^d)*10^10 s(2^d) + t(2^d)
== ==================== ============ ====================
20 0.237251764919808326 115.34589710 0.237251776454398036
21 0.237251771317612979 52.59702970 0.237251776577315949
22 0.237251774173347724 24.08221952 0.237251776581569676
23 0.237251775469086555 11.06766714 0.237251776575853269
24 0.237251776066813995 5.10395459 0.237251776577209454
25 0.237251776340760021 2.36119196 0.237251776576879217
26 0.237251776467109357 1.09553336 0.237251776576662693
27 0.237251776525743797 0.50967952 0.237251776576711749
28 0.237251776552887645 0.23771866 0.237251776576659511
29 0.237251776565549906 0.11113468 0.237251776576663374
30 0.237251776571456873 0.05207020 0.237251776576663892
31 0.237251776574218065 0.02444677 0.237251776576662742
32 0.237251776575513036 0.01149984 0.237251776576663020
33 0.237251776576121140 0.00541938 0.237251776576663078
.
which agrees with all the terms in the Data section and suggests likely values for additional terms.
(End)
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