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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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Cited by 13 (1 self)
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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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Zeroes of Dirichlet Lfunctions and irregularities in the distribution of primes
 Math. Comp. S
, 1999
"... Abstract. Seven widely spaced regions of integers with π4,3(x) <π4,1(x) have been discovered using conventional prime sieves. Assuming the generalized Riemann hypothesis, we modify a result of Davenport in a way suggested by the recent work of Rubinstein and Sarnak to prove a theorem which makes ..."
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Cited by 4 (2 self)
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Abstract. Seven widely spaced regions of integers with π4,3(x) <π4,1(x) have been discovered using conventional prime sieves. Assuming the generalized Riemann hypothesis, we modify a result of Davenport in a way suggested by the recent work of Rubinstein and Sarnak to prove a theorem which makes it possible to compute the entire distribution of π4,3(x) − π4,1(x) including the sign change (axis crossing) regions, in time linear in x, using zeroes of L(s, χ),χ the nonprincipal character modulo 4, generously provided to us by Robert Rumely. The accuracy with which the zeroes duplicate the distribution (Figure 1) is very satisfying. The program discovers all known axis crossing regions and finds probable regions up to 10 1000. Our result is applicable to a wide variety of problems in comparative prime number theory. For example, our theorem makes it possible in a few minutes of computer time to compute and plot a characteristic sample of the difference li(x) − π(x) with fine resolution out to and beyond the region in the vicinity of 6.658 × 10 370 discovered by te Riele. This region will be analyzed elsewhere in conjunction with a proof that there is an earlier sign change in the vicinity of 1.39822 × 10 316. 1.
THE PRIME NUMBER RACE AND ZEROS OF LFUNCTIONS OFF THE CRITICAL LINE
"... We examine the effects of certain hypothetical configurations of zeros of Dirichlet Lfunctions lying off the critical line on the distribution of primes in arithmetic progressions. 1. ..."
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Cited by 1 (1 self)
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We examine the effects of certain hypothetical configurations of zeros of Dirichlet Lfunctions lying off the critical line on the distribution of primes in arithmetic progressions. 1.
Chebyshev's Bias
, 1994
"... this paper we take a somewhat different point of view in our attempt to analyze Chebyshev's phenomenon and its generalizations, which we call "Chebyshev's bias". Our purpose has been to examine these issues both theoretically and numerically and, in particular, to give numerical ..."
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this paper we take a somewhat different point of view in our attempt to analyze Chebyshev's phenomenon and its generalizations, which we call "Chebyshev's bias". Our purpose has been to examine these issues both theoretically and numerically and, in particular, to give numerical values to these biases. Let a 1 ; a 2 ; : : : ; a r 2 A q be distinct, and define
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