The integrant of the prime gap distribution function is given by :
We substitute u = 1/log(t) and after removing a factor of u^2 we are left with a polynomial of degree N-1 :
with N being defined by A(r,N) ≠ 0 and A(r,k) = 0 for all k > N (Table of N as function of r)
The real-valued roots of this polynomial are the locations of the minima and maxima of the prime gap distribution integral and were evaluated with the positive coefficients A(r,k) known to 50 siginificant (decimal) places and all calculations to find all roots (real and complex-conjugated) and subsequent evaluation of the distribution integral were conducted with the multi-precision arithmetic library gmp to the same precision.
The following table shows the location of the roots ( in terms of the substitution variable u and then in terms of the integration variable t ) for the case of r=80 for which the polynomial is of degree 33 (N=34). In the third column the value of the distribution integral is given ( with lower integration limit a=2 ) but from which a constant value called offset is subtracted. Its value is approximately -1.738574e+18 which was chosen to give exactly zero in the third column at the position of the last root. This choice demonstrates a remarkable feature of L(r,x)-offset namely that its first minimum of -1.933135e+14 is followed by a maximum of 8.392811e+03 (10 orders of magnitude smaller ) which are later followed by a series of minima and maxima whose values are equal to each other to within 1.3e-13 which is 30 orders of magnitudesmaller than the value of L(r,x) at the location of the last root. This behavior is reminiscent to the Gibb's phenomenon occuring in the Fourier series expansions of discontinuous functions.
| Location of root | L(r,x) - offset | |
|---|---|---|
| u | t | |
| 1.196782e+00 | 2.306137e+00 | -1.933135e+14 |
| 6.112930e-01 | 5.133957e+00 | 8.392811e+03 |
| 5.009256e-01 | 7.361801e+00 | -5.231697e+00 |
| 4.171302e-01 | 1.099381e+01 | 5.030219e-03 |
| 3.598720e-01 | 1.609913e+01 | -1.384545e-05 |
| 3.209839e-01 | 2.254291e+01 | 1.231797e-07 |
| 2.945499e-01 | 2.981495e+01 | -6.955581e-09 |
| 2.680114e-01 | 4.172853e+01 | 3.015965e-10 |
| 2.441798e-01 | 6.005991e+01 | -1.102101e-11 |
| 2.245537e-01 | 8.590802e+01 | 6.306679e-13 |
| 2.081558e-01 | 1.220089e+02 | 1.032041e-13 |
| 1.925737e-01 | 1.799749e+02 | 1.280941e-13 |
| 1.917547e-01 | 1.840109e+02 | 1.280929e-13 |
| 1.833075e-01 | 2.339984e+02 | 1.285066e-13 |
| 1.725226e-01 | 3.290933e+02 | 1.277254e-13 |
| 1.633614e-01 | 4.554999e+02 | 1.280688e-13 |
| 1.553123e-01 | 6.255567e+02 | 1.279142e-13 |
| 1.481632e-01 | 8.534744e+02 | 1.279952e-13 |
| 1.417650e-01 | 1.157393e+03 | 1.279436e-13 |
| 1.359951e-01 | 1.561192e+03 | 1.279846e-13 |
| 1.307457e-01 | 2.097360e+03 | 1.279429e-13 |
| 1.259216e-01 | 2.811423e+03 | 1.279988e-13 |
| 1.214386e-01 | 3.769175e+03 | 1.278971e-13 |
| 1.172176e-01 | 5.070215e+03 | 1.281603e-13 |
| 1.131730e-01 | 6.877636e+03 | 1.271138e-13 |
| 1.091846e-01 | 9.497658e+03 | 1.345189e-13 |
| 1.049928e-01 | 1.369057e+04 | 0.000000e+00 |
As the value of r is decreased the above mentioned trends become less pronounced until at r < 10 the number of roots drops below 5.
In summary, the present investigations show the L(r,x) shows at larger values of the half-gap size unreasonable behavior at small values of x followed by a plateau of almost constant values which becomes larger and flatter as r increases. This allows to choose from a wide range of values for the lower integration limit a without any consequence for the actual prime gap count. It is observered that the value of x=2*r falls securely into the area of the plateau at all values of r and we therefore suggest to take the value of 2*r as the lower integration limit.