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Prime Number Races
 Amer. Math. Monthly
"... 1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that ..."
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1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking
Biases in the . . .
, 1999
"... Rubinstein and Sarnak investigated systems of inequalities of the form π(x;q,a1)> · · ·> π(x;q,ar), 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 Lfunctions mod q, that the set of positive ..."
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Rubinstein and Sarnak investigated systems of inequalities of the form π(x;q,a1)> · · ·> π(x;q,ar), 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 Lfunctions 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 ai in general, even if the ai 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 δq;a1,...,ar themselves vary under permutations of the ai. In this paper, 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