Biases in the Shanks-Renyi Prime Number Race
Rubinstein and Sarnak investigated systems of inequalities of the form pi(x;q,a_1) > ... > pi(x;q,a_r), where pi(x;q,a) denotes the number of primes up to x that are congruent to a mod q. They showed, under standard hypotheses on the zeros of Dirichlet L-functions mod q, that the set of positive real numbers x for which these inequalities hold has positive (logarithmic) density delta_{q;a_1,dots,a_r} > 0. They also discovered the surprising fact that a certain distribution associated with these densities is not symmetric under permutations of the residue classes a_j in general, even if the a_j are all squares or all nonsquares mod q (a condition necessary to avoid obvious biases of the type first observed by Chebyshev). This asymmetry suggests, contrary to prior expectations, that the densities delta_{q;a_1,dots,a_r} themselves vary under permutations of the a_j. In this paper, we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities delta_{q;a_1,dots,a_r}, and we use this formula to calculate many of these densities when q <= 12 and r <= 4. For the special moduli q = 8 and q = 12, and for {a_1,a_2,a_3} a permutation of the nonsquares {3,5,7} mod 8 and {5,7,11} mod 12, respectively, we rigorously bound the error in our calculations, thus verifying that these densities are indeed asymmetric under permutation of the a_j. We also determine several situations in which the densities delta_{q;a_1,dots,a_r} remain unchanged under certain permutations of the a_j, and some situations in which they are provably different.