标题：Numerical and graphical description of all axis crossing regions for moduli <mml:math alttext="$4$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>4</mml:mn></mml:math> and <mml:math alttext="$8$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>8</mml:mn></mml:math> which occur befor <mml:math alttext="$10^{12}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:math>
期刊名称：International Journal of Mathematics and Mathematical Sciences
印刷版ISSN：0161-1712
电子版ISSN：1687-0425
出版年度：1979
卷号：2
DOI：10.1155/S0161171279000119
出版社：Hindawi Publishing Corporation
摘要：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).
