BIASES IN THE SHANKS–RÉNYI PRIME NUMBER RACE

ANDREY FEUERVERGER AND GREG MARTIN

Abstract. Rubinstein and Sarnak investigated systems of inequalities of the form π(x; q, a_{1}) > ⋯ > π(x; q, a_{r}), 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 δ_{q;a_{1},··· ,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 δ_{q;a_{1},··· ,a_{r}} themselves vary under permutations of the a_{j}.

Here we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities δ_{q;a_{1},··· ,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 δ_{q;a_{1},··· ,a_{r}} remain unchanged under certain permutations of the a_{j}, and some situations in which they are provably different.