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Martin2000

G. Martin, Asymmetries in the Shanks-Rényi prime number race.  In "Number theory for the millennium, II (Urbana, IL, 2000)," A K Peters, Natick, MA, 2002.  Preprint.  MR 1 956 261

Abstract: 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.