Biases in the shanks-rénvi prime number race

Abstract

Investigated systems of inequalities of the form π(x; q, a1) > … > π(x; q, ar), where π(x; q, b) denotes the number of primes up to x that are congruent to b 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 δq;a1, …, ar. > 0 They also discovered the surprising fact that a certain distribution associated with these densities is not symmetric under permutations of the residue classes aj in general, even if the a, are all squares or all nonsquares mod q (a condition necessary to avoid obvious biases of the type first observed by Cheby- shev). This asymmetry suggests, contrary to prior expectations, that the densities δq;a1, …, ar, themselves vary under permutations of the aj. Here we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities δq;a1, …, ar, 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 {a1, a2, a3} 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 aj. We also determine several situations in which the densities δq;a1, …, ar remain unchanged under certain permutations of the aj, and some situations in which they are provably different. © A K Peters, Ltd.