Department of Mathematics and Computer Science, University of South Carolina, Columbia 29208, S.C., USA
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.
Let denote the number of primes and , and for positive integers let . Negative values of less than occur in six widely spaced regions. The first three regions, investigated by Leech , Shanks  and Lehmer [6 ], contain only a few thousand negative values of . However, the authors  have recently discovered new regions, the sixth occurring before billion and containing more than half a billion negative values of . In this paper numerical and graphical details of all six regions are given. Moreover, new results for the modulus are presented. Previously, no negative values have been found for , , or and our search to reveals no such values for or . For we have discovered the first two regions of negative values. The first of these regions, beginning at , contains negative values of ; the second occurs in the vicinity of billion and contains more than a billion negative values of .