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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
Prime number races and zeros of Dirichlet Lfunctions
, 2009
"... This Research in Teams meeting focused on the finer behaviour of the function π(x; q, a), which denotes the number of prime numbers of the form qn + a that are less than or equal to x. Dirichlet’s famous theorem on primes in arithmetic progressions asserts that that there are infinitely many primes ..."
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This Research in Teams meeting focused on the finer behaviour of the function π(x; q, a), which denotes the number of prime numbers of the form qn + a that are less than or equal to x. Dirichlet’s famous theorem on primes in arithmetic progressions asserts that that there are infinitely many primes of the form qn + a when a is a reduced residue modulo q (that is, when a and q are relatively prime), and so π(x; q, a) is unbounded. If a and b are reduced residues modulo q, then we may ask whether the inequality π(x; q, a)> π(x; q, b) (1) is satisfied for arbitrarily large x. Chebyshev observed that for the triple (q; a, b) = (4; 3, 1), the inequality (1) holds for all small x. In fact, he asked whether this inequality would continue to hold for all x. However, in 1914 Littlewood proved that for each of the triples (4; 1, 3) and (4; 3, 1), there are arbitrarily large values of x such that the inequality (1) holds. These inequalities can be thought of as a “prime number race ” between two contestants, Team 1 and Team 3. In these terms, Chebyshev observed that Team 3 usually leads Team 1; Littlewood’s theorem asserts that each team takes turns leading the prime number race infinitely often. Over the years, researchers have attempted to prove that there are triples (q; a, b) such that (1) holds for arbitrarily large x. However, only a few such results have been established. For many triples (q; a, b) with values of q ranging up to 100, it is known that the inequality (1) holds for arbitrarily large x; these results,