Moments for primes in arithmetic progressions, I

R. C. Vaughan

doi:10.1215/S0012-7094-03-12026-8

Moments for primes in arithmetic progressions, I

R. C. Vaughan

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

The second moment ∑_q≤Q ∑_a=1^q (ψ(x; q, a) - ρ (x; q, a))² is investigated with the novel approximation ρ(x; q, a) = ∑_{n≤x n≡a (mod q)} F _R(n), where F_R(n) = ∑_r≤R μ(r)/φ(r) ∑_{b=1 (b,r)=1}^r e(bn/r), and it is shown that when R ≤ log^A x, this leads to more precise conclusions than in the classical Montgomery-Hooley case.

Original language	English (US)
Pages (from-to)	371-383
Number of pages	13
Journal	Duke Mathematical Journal
Volume	120
Issue number	2
DOIs	https://doi.org/10.1215/S0012-7094-03-12026-8
State	Published - Nov 1 2003

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1215/S0012-7094-03-12026-8

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abstract = "The second moment ∑q≤Q ∑a=1q (ψ(x; q, a) - ρ (x; q, a))2 is investigated with the novel approximation ρ(x; q, a) = ∑n≤x n≡a (mod q) F R(n), where FR(n) = ∑r≤R μ(r)/φ(r) ∑b=1 (b,r)=1r e(bn/r), and it is shown that when R ≤ logA x, this leads to more precise conclusions than in the classical Montgomery-Hooley case.",

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N2 - The second moment ∑q≤Q ∑a=1q (ψ(x; q, a) - ρ (x; q, a))2 is investigated with the novel approximation ρ(x; q, a) = ∑n≤x n≡a (mod q) F R(n), where FR(n) = ∑r≤R μ(r)/φ(r) ∑b=1 (b,r)=1r e(bn/r), and it is shown that when R ≤ logA x, this leads to more precise conclusions than in the classical Montgomery-Hooley case.

AB - The second moment ∑q≤Q ∑a=1q (ψ(x; q, a) - ρ (x; q, a))2 is investigated with the novel approximation ρ(x; q, a) = ∑n≤x n≡a (mod q) F R(n), where FR(n) = ∑r≤R μ(r)/φ(r) ∑b=1 (b,r)=1r e(bn/r), and it is shown that when R ≤ logA x, this leads to more precise conclusions than in the classical Montgomery-Hooley case.

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JO - Duke Mathematical Journal

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Moments for primes in arithmetic progressions, I

Abstract

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