On October 1st, 2017 our “4000 Chebyshev Bias Tester” completed testing all primes up to 10*1012 and found the 8th sign-changing zone for ∆{4;3,1}(x), predicted in 2001 by American mathematicians Carter Bays, Kevin Ford, Richard H. Hudson and Michael Rubinstein to be around 9.318*1012 .
In 2004 these predictions were confirmed by a group of French mathematicians Marc Deléglise, Pierre Dusart, and Xavier-François Roblot. They also predicted the next 9th sign-changing zone to exist around 9.97*1017.
The 8th sign-changing zone, found by us, starts from 9103362505801 and ends with 9543313015309 (the first and the last primes where value of ∆{4;3,1}(x) equals to -1).
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 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. Since then the main efforts of mathematicians were concentrated on finding the theoretical explanation to this phenomenon as well as on predicting and testing the new sign-changing zones for divisors other than 4.
We plan to publish our main results in the nearest future.