Hardy Littlewood Constant C

Since for Goldbach for Gaussian integers, primes of the form n2+1 matter, we got interested in them. We also looked at the constant C defined by Hardy-Littlewood, which gives the ratio between primes on the row Im(z)=1 and the primes on the row Im(z)=0. We computed the ratio
| { z = a + 1 i  |   z is Gaussian prime ,  a ≤ n } |
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| { z = a + 0 i  |   z is Gaussian prime ,  a ≤ n } |
Up to n=157819000000 = 1.5 * 1011, we see the fraction is 4378609531/3189494198 = 1.37282 ... In the end, we had to check primes of the order n2 which is 2.5 * 1022. The program can not be simpler. It is a ``one liner". (But the trick is to get to 1012 but we hope we will get there eventually ...)
n=m=0;Do[If[PrimeQ[k^2+1],m++];If[PrimeQ[k]&&Mod[k,4]==3,n++;Print[N[m/n,20]]],{k,10^12}]
You see that the convergence is slow. The fluctuations are still of the order 10-5. Thats about the same order than what was got 100 years ago or what Shanks and Wunderlich confirmed. Wunderlich went to n=14'000'000 = 1.4 * 107. This means we go 10'000 higher and have to check primes which are 108 times larger. Wunderlich wrote his paper in 1973 which is 43 years. This illustrates well Moore's law on hardware advancement!