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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
Zeros of Dirichlet LFunctions near the Real Axis and Chebyshev's Bias
 JOURNAL OF NUMBER THEORY
, 2001
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Asymmetries in the Shanks–Rényi
, 2000
"... Abstract. It has been wellobserved that an inequality of the type π(x; q, a)> π(x; q, b) is more likely to hold if a is a nonsquare modulo q and b is a square modulo q (the socalled “Chebyshev Bias”). For instance, each of π(x; 8, 3), π(x; 8, 5), and π(x; 8, 7) tends to be somewhat larger than π(x ..."
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Abstract. It has been wellobserved that an inequality of the type π(x; q, a)> π(x; q, b) is more likely to hold if a is a nonsquare modulo q and b is a square modulo q (the socalled “Chebyshev Bias”). For instance, each of π(x; 8, 3), π(x; 8, 5), and π(x; 8, 7) tends to be somewhat larger than π(x; 8, 1). However, it has come to light that the tendencies of these three π(x; 8, a) to dominate π(x; 8, 1) have different strengths. A related phenomenon is that the six possible inequalities of the form π(x; 8, a1)> π(x; 8, a2)> π(x; 8, a3) with {a1, a2, a3} = {3, 5, 7} are not all equally likely—some orderings are preferred over others. In this paper we discuss these phenomena, focusing on the moduli q = 8 and q = 12, and we explain why the observed asymmetries (as opposed to other possible asymmetries) occur. 1. Background Let π(x; q, a) denote the number of primes not exceeding x that are congruent to a modulo q. We have known since the work of Dirichlet that the two counting functions π(x; q, a) and π(x; q, b) are asymptotically equal as x tends to infinity (as long as a and b are both coprime to q). However, more complicated behavior emerges when we compare these counting functions for finite values of x. Imagine π(x; q, a) and π(x; q, b) as representing the