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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
LIMITING DISTRIBUTIONS OF THE CLASSICAL ERROR TERMS OF PRIME NUMBER THEORY
, 1306
"... ABSTRACT. Let φ: [0,∞) → R and let y0 be a nonnegative constant. Let (λn)n∈N be a nondecreasing sequence of positive numbers which tends to infinity, let(rn)n∈N be a complex sequence, andcareal number. Assume thatφis squareintegrable on[0,y0] and fory ≥ y0,φcan be expressed as φ(y) = c+ℜ for any ..."
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Cited by 1 (0 self)
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ABSTRACT. Let φ: [0,∞) → R and let y0 be a nonnegative constant. Let (λn)n∈N be a nondecreasing sequence of positive numbers which tends to infinity, let(rn)n∈N be a complex sequence, andcareal number. Assume thatφis squareintegrable on[0,y0] and fory ≥ y0,φcan be expressed as φ(y) = c+ℜ for anyX ≥ X0> 0 whereE(y,X) satisfies 1 lim Y→ ∞ Y λn≤X rne iλny)
P (3> 1mod4)=0.9959280...,
, 2006
"... How do we quantify irregularities in the distribution of prime numbers? Define where gcd(a, q) = 1. A wellknown result: πq,a(n) =#{p ≤ n: p ≡ a mod q} ln(n) lim n→ ∞ n πq,a(n) = 1 ϕ(q) informs us that primes are asymptotically equidistributed modulo q, whereϕ(q) is the Euler totient. There is, ho ..."
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How do we quantify irregularities in the distribution of prime numbers? Define where gcd(a, q) = 1. A wellknown result: πq,a(n) =#{p ≤ n: p ≡ a mod q} ln(n) lim n→ ∞ n πq,a(n) = 1 ϕ(q) informs us that primes are asymptotically equidistributed modulo q, whereϕ(q) is the Euler totient. There is, however, unrest beneath the surface of such symmetry. For fixed a1, a1,..., ar and q, define
COUNTING PRIMES IN RESIDUE CLASSES
"... Abstract. We explain how the MeisselLehmerLagariasMillerOdlyzko method for computing π(x) can be used to compute efficiently π(x, k, l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n ± 1 less than x for several val ..."
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Abstract. We explain how the MeisselLehmerLagariasMillerOdlyzko method for computing π(x) can be used to compute efficiently π(x, k, l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n ± 1 less than x for several values of x up to 1020 and found a new region where π(x, 4, 3) is less than π(x, 4, 1) near x = 1018. hal00863138, version 1 19 Sep 2013 1.
Quadratic Dirichlet LSeries
, 2005
"... Let D = 1 or D be a fundamental discriminant [1]. The KroneckerJacobiLegendre symbol (D/n) is a completely multiplicative function on the positive integers: where n = p e1 1 p e2 2 · · · p ek k D ..."
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Let D = 1 or D be a fundamental discriminant [1]. The KroneckerJacobiLegendre symbol (D/n) is a completely multiplicative function on the positive integers: where n = p e1 1 p e2 2 · · · p ek k D
D p
, 2005
"... Let D =1or D be a fundamental discriminant [1]. The KroneckerJacobiLegendre symbol (D/n) is a completely multiplicative function on the positive integers: µ ¶ ⎪⎨ kY µ ¶ ej D D if n ≥ 2, = pj n ⎪⎩ j=1 1 if n =1 where n = p e1 1 p e2 2 ···p ek k µ ¶ ..."
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Let D =1or D be a fundamental discriminant [1]. The KroneckerJacobiLegendre symbol (D/n) is a completely multiplicative function on the positive integers: µ ¶ ⎪⎨ kY µ ¶ ej D D if n ≥ 2, = pj n ⎪⎩ j=1 1 if n =1 where n = p e1 1 p e2 2 ···p ek k µ ¶