Biases in the Shanks—Rényi Prime Number Race

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, 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 δ q;a 1 ,...,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 a i in general, even if the a i 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 ,...,ar themselves vary under permutations of the a i .In this paper, we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities δ q;a 1 ,...,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 {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 i . We also determine several situations in which the densities δ q;a 1 ,...,ar remain unchanged under certain permutations of the a i , and some situations in which they are provably different.

Chebyshevʼs bias in Galois extensions of global function fields

“…Then, lim c→0 + G(t, c) = π by the dominated convergence theorem (see p. 548 of [6]). Now, we claim thatρ(−ξ ) =ρ(ξ ) for all ξ .…”

“…The computation of δ comes from the (easiest) special case (r = 2) of [4, Theorem 2.1], which is the function field reformulation of the corresponding result in [6]. As in [6], we define the measure…”

Biases in the prime number race of function fields

“…The aim of our paper is to give a function field version of the aforementioned formula of Feuerverger and Martin in [3]. It is probably not surprising that most of our proof can be obtained by closely following the strategy of [3] because of the strong resemblance between the main results in [4] and [2].…”

“…It is probably not surprising that most of our proof can be obtained by closely following the strategy of [3] because of the strong resemblance between the main results in [4] and [2]. Our contribution in this paper is to adapt the methods of [3] in order to deal with the problem that the Fourier transform of a certain probability measure ρ, unlike the number field case, does not decay fast enough at infinity. This problem is present only in the function field case, due to the fact that there are only finitely many zeta zeros.…”

“…One of the most important work on this topic is a paper [4] of Rubinstein and Sarnak, which provided justification of this bias under certain very plausible hypotheses. Building up on this, Feuerverger and Martin [3] derived quite a general formula, (2.54) in Theorem 4 of [3], which expresses the numerical value of the bias in a prime number race among residue classes in terms of certain explicit integrals. Their formula enables one to deduce the existence of various symmetries of the bias under permutations on the residue classes.…”