# Hardy Littlewood Constant C

Since for Goldbach for Gaussian integers, primes of the form n^{2}+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 } | -------------------------------------------------------- | { z = a + 0 i | z is Gaussian prime , a ≤ n } |Up to n=157819000000 = 1.5 * 10

^{11}, we see the fraction is 4378609531/3189494198 = 1.37282 ... In the end, we had to check primes of the order n

^{2}which is 2.5 * 10

^{22}. The program can not be simpler. It is a ``one liner". (But the trick is to get to 10

^{12}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 * 10

^{7}. This means we go 10'000 higher and have to check primes which are 10

^{8}times larger. Wunderlich wrote his paper in 1973 which is 43 years. This illustrates well Moore's law on hardware advancement!