In 1853 Chebyshev observed that there were more primes of a form 4n+3 than the primes of a form 4n+1. It was proven that Chebyshev’s bias is closely related to the Generalized Riemann Hypothesis and zeros of Dirichlet L-function. Our objective was to expand 1000 times the 10^12 range tested between 1957 and 1996 and to find new sign-changing zones for 15 “most biased prime number races” established in 2013. As a result, we found 25 new zones between 10^12 and 10^15, that compared with only 11 zones known up to 10^12. Out of all 25 zones, 4 are the first found and present the most scientific interest. Some of the small and narrow zones happened to be much lower than was predicted by theoretical models. Therefore, these models need revision. All 18 sequences in total are published with OEIS.


Andrey Sergeevich Shchebetov