Biases in the Shanks-Rényi Prime Number Race
by
Greg Martin
University of Toronto
Coauthors: Andrey Feuerverger

Let \pi(x;q, a) denote the number of primes not exceeding x that are congruent to a mod q, and let \delta_{q;a1, ..., ar} be the asymptotic density (suitably defined) of the set of positive real numbers x such that \pi(x;q, a₁) > \pi(x;q, a₂) > ... > \pi(x;q, a_r). From observations (beginning with Chebyshev) it seems that pi(x;q, a) tends to exceed \pi(x;q, b) when a is a nonsquare mod q and b is a square mod q, and Rubinstein and Sarnak proved (conditionally) that \delta_{q;a, b} > 1/2 if and only if a is a nonsquare and b is a square mod q. In this talk, we describe computations establishing (conditionally) the values of the densities \delta_{q;a1, ..., ar} for several small moduli q; in particular, we show that these densities do depend on the ordering of the a_j, even when the a_j are all squares or all nonsquares mod q. We also establish some situations in which the densities \delta_{q;a1, ..., ar} are provably equal under certain permutations of the a_j, and some other situations in which certain inequalities can be established.

Greg Martin - Manuscripts

Date received: January 10, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadx-09.