2000/10/09 by Greg Martin, Martin, Greg
Mathematics · #11N15 (11N69) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N15
paper · pdf · doi:10.48550/arxiv.math/0010086
11 pages; submitted to the conference proceedings of the Millennial Conference on Number Theory (University of Illinois at Urbana-Champaign, 2000). Minor revisions only
arxiv created 2000/11/17 · arxiv updated 2009/11/30
It has been well-observed that an inequality of the type π(x;q,a) > π(x;q,b) is more likely to hold if a is a non-square modulo q and b is a square modulo q (the so-called ``Chebyshev Bias''). For instance, each of π(x;8,3), π(x;8,5), and π(x;8,7) tends to be somewhat larger than π(x;8,1). However, it has come to light that the tendencies of these three π(x;8,a) to dominate π(x;8,1) have different strengths. A related phenomenon is that the six possible inequalities of the form π(x;8,a1) > π(x;8,a2) > π(x;8,a3) with \a1,a2,a3\=\3,5,7\ are not all equally likely---some orderings are preferred over others. In this paper we discuss these phenomena, focusing on the moduli q=8 and q=12, and we explain why the observed asymmetries (as opposed to other possible asymmetries) occur.