Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur befor 10 12

Let π b , c ( x ) denote the number of primes ≤ x and ≡ c ( mod b ) , and for positive integers x let Δ b ( x , c , l ) = π b , c ( x ) − π b , l ( x ) . Negative values of Δ 4 ( x , 3 , 1 ) less than 10 12 occur in six widely spaced regions. The first three regions, investigated by Leech [6], Shanks [9] and Lehmer [6 ], contain only a few thousand negative values of Δ 4 ( x , 3 , 1 ) . However, the authors [1] have recently discovered 3 new regions, the sixth occurring before 20 billion and containing more than half a billion negative values of Δ 4 ( x , 3 , 1 ) . In this paper numerical and graphical details of all six regions are given. Moreover, new results for the modulus 8 are presented. Previously, no negative values have been found for Δ 8 ( x , c , 1 ) , c = 3 , 5 , or 7 and our search to 10 12 reveals no such values for Δ 8 ( x , 3 , 1 ) or Δ 8 ( x , 7 , 1 ) . For Δ 8 ( x , 5 , 1 ) we have discovered the first two regions of negative values. The first of these regions, beginning at x = 588067889 , contains 422 , 500 negative values of Δ 8 ( x , 5 , 1 ) ; the second occurs in the vicinity of 35 billion and contains more than a billion negative values of Δ 8 ( x , 5 , 1 ) .