On October 13, 2017 we published in Online Encyclopedia of Integer Sequences (OEIS) under  A007350 the full data on all 194367 primes where ∆{4;3,1}(x) changes sign. Staring from 12502nd term equal to 9103362505801 the published sequence includes data on the 8th sign-changing zone, that was predicted back in 2001 by a group of American mathematicians, but was never actually found.

The function ∆{4;3,1}(x) = π{4,3}(x) – π{4,1}(x) represents the difference between primes  of the form 4k + 3 and primes of the form 4k + 1 up to a given x and plays an important role in number theory. In 1853 the brilliant Russian mathematician Pafnuty Lvovich Chebyshev observed that this functions becomes negative quite rarely. Since then this phenomenon is known all over the world as Chebyshev’s bias. The direct numerical testing of this function present a complex and daunting computational problem complicated by inefficient algorithms, lack of memory and other limitations of modern computers.

The first seven sign-changing zones of this function were found between 1957 and 2001. The full data sets related to them has never been published before (e.g., OEIS  A007350 contained just 301 terms prior to our publication). The 8th sign-changing zone was predicted back in 2001, but never actually found.

All our data available for download from OEIS as well as from our repository.

We plan to continue publishing our results on Chebyshev’s bias (including data on the narrow 9th sign-changing zone for ∆{4;3,1}(x), discovered by us) in the near future.