Inequities in the Shanks-Renyi Prime Number Race
by
Greg Martin
University of British Columbia

It has been well-observed that an inequality of the type \pi(x;q, a) > \pi(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'' in comparative prime number theory).  However, it has come to light that the tendencies of the various \pi(x;q, a) (for nonsquares a) to dominate \pi(x;q, b) have different strengths. A related phenomenon is that the six possible inequalities of the form \pi(x;q, a₁) >  \pi(x;q, a₂) >  \pi(x;q, a₃), with a₁, a₂, a₃ all non-squares modulo q, are not all equally likely; some orderings are preferred over others. For given values q, a, b, ..., these tendencies can be quantified and computed, but only using laborious numerical integration of functions involving zeros of the appropriate Dirichlet L-functions. In this talk, we present a framework for explaining which nonsquares a are most dominant for a given square b, for example, based only on elementary properties of the congruence classes a modulo q, rather than the complicated computations just mentioned.

Date received: April 27, 2004