Numerical and graphical description of all axis crossing regions for moduli4and8which occur befor1012

Abstract

Letπb,c(x)denote the number of primes≤xand≡c(modb), and for positive integersxletΔb(x,c,l)=πb,c(x)−πb,l(x). Negative values ofΔ4(x,3,1)less than1012occur 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 ofΔ4(x,3,1). However, the authors [1] have recently discovered3new regions, the sixth occurring before20billion and containing more than half a billion negative values ofΔ4(x,3,1). In this paper numerical and graphical details of all six regions are given. Moreover, new results for the modulus8are presented. Previously, no negative values have been found forΔ8(x,c,1),c=3,5, or7and our search to1012reveals no such values forΔ8(x,3,1)orΔ8(x,7,1). ForΔ8(x,5,1)we have discovered the first two regions of negative values. The first of these regions, beginning atx=588067889, contains422,500negative values ofΔ8(x,5,1); the second occurs in the vicinity of35billion and contains more than a billion negative values ofΔ8(x,5,1).