Full data on all 8 zones with 0 values for ∆{4;3,1}(x) in Chebyshev’s bias published

Оn October 15, 2017 we published in Online Encyclopedia of Integer Sequences (OEIS) under  A007351 the full data on all 419467 primes where ∆{4;3,1}(x) is equal to zero. Staring from 27410th term equal to 9103362505753 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  A007351 contained just 1000 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 make available the data on the narrow 9th sign-changing zone also found by us later.

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.