Carter Bays and Richard H. Hudson

Copyright © 1979 Carter Bays and Richard H. Hudson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let πb,c(x) denote the number of primes ≤x and ≡c(modb), and for positive integers x let Δb(x,c,l)=πb,c(x)−πb,l(x). Negative values of Δ4(x,3,1) less than 1012 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 Δ4(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 Δ4(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 Δ8(x,c,1), c=3,5, or 7 and our search to 1012 reveals no such values for Δ8(x,3,1) or Δ8(x,7,1). For Δ8(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 Δ8(x,5,1); the second occurs in the vicinity of 35 billion and contains more than a billion negative values of Δ8(x,5,1).