Sieve Methods, Exponential Sums, and their Applications in Number Theory

G. R. H. Greaves; G. Harman; M. N. Huxley

doi:10.1017/CBO9780511526091

This volume comprises the proceedings of the 1995 Cardiff symposium on sieve methods, exponential sums, and their applications in number theory. Included are contributions from many leading international figures in this area which encompasses the main branches of analytic number theory. In particular, many of the papers reflect the interaction between the different fields of sieve theory, Dirichlet series (including the Riemann Zeta-function), and exponential sums, whilst displaying the subtle interplay between the additive and multiplicative aspects of the subjects. The fundamental problems discussed include recent work on Waring's problem, primes in arithmetical progressions, Goldbach numbers in short intervals, the ABC conjecture, and the moments of the Riemann Zeta-function.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Foreword
pp vii-vii
- By G. R. H. Greaves, G. J. Harman, M. N. Huxley
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Index of Authors
pp viii-x
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Participants in the Symposium
pp xi-xii
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1 - The Exceptional Set for Goldbach's Problem in Short Intervals
pp 1-54
- By R. C. Baker, G. Harman, J. Pintz
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2 - On an Additive Property of Stable Sets
pp 55-64
- By A. Balog, I. Z. Ruzsa
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3 - Squarefree Values of Polynomials and the abc-Conjecture
pp 65-86
- By J. Browkin, M. Filaseta, G. Greaves, A. Schinzel
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4 - The Values of Binary Linear Forms at Prime Arguments
pp 87-100
- By J. Brüdern, R. J. Cook, A. Perelli
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5 - Some Applications of Sieves of Dimension exceeding 1
pp 101-108
- By H. Diamond, H. Halberstam
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6 - Representations by the Determinant and Mean Values of L-Functions
pp 109-116
- By W. Duke, J. Friedlander, H. Iwaniec
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7 - On the Montgomery-Hooley Asymptotic Formula
pp 117-142
- By D. A. Goldston, R. C. Vaughan
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8 - Franel Integrals
pp 143-160
- By R.R. Hall
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9 - Eratosthenes, Legendre, Vinogradov and beyond
pp 161-174
- By G. J. Harman
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10 - On Hypothesis K* in Waring's Problem
pp 175-186
- By C. Hooley
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11 - Moments of Differences between Square-free Numbers
pp 187-204
- By M. N. Huxley
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12 - On the Ternary Additive Divisor Problem and the Sixth Moment of the Zeta-Function
pp 205-244
- By A. Ivić
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13 - A Variant of the Circle Method
pp 245-254
- By M. Jutila
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14 - The Resemblance of the Behaviour of the Remainder Terms Eσ(t), Δ1-2σ(x) and R(σ + it)
pp 255-274
- By I. Kiuchi, K. Matsumoto
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15 - A Note on the Number of Divisors of Quadratic Polynomials
pp 275-282
- By J. McKee
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16 - On the Distribution of Integer Points in the Real Locus of an Affine Toric Variety
pp 283-292
- By B. Z. Moroz
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17 - An Asymptotic Expansion of the Square of the Riemann Zeta-Function
pp 293-308
- By Y. Motohashi
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18 - The Mean Square of Dedekind Zeta-Functions of Quadratic Number Fields
pp 309-324
- By Y. Motohashi
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19 - Artin's Conjecture and Elliptic Analogues
pp 325-344
- By M. R. Murty
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Sieve Methods, Exponential Sums, and their Applications in Number Theory

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Foreword
pp vii-vii

Index of Authors
pp viii-x

Participants in the Symposium
pp xi-xii

1 - The Exceptional Set for Goldbach's Problem in Short Intervals
pp 1-54

2 - On an Additive Property of Stable Sets
pp 55-64

3 - Squarefree Values of Polynomials and the abc-Conjecture
pp 65-86

4 - The Values of Binary Linear Forms at Prime Arguments
pp 87-100

5 - Some Applications of Sieves of Dimension exceeding 1
pp 101-108

6 - Representations by the Determinant and Mean Values of L-Functions
pp 109-116

7 - On the Montgomery-Hooley Asymptotic Formula
pp 117-142

8 - Franel Integrals
pp 143-160

9 - Eratosthenes, Legendre, Vinogradov and beyond
pp 161-174

10 - On Hypothesis K* in Waring's Problem
pp 175-186

11 - Moments of Differences between Square-free Numbers
pp 187-204

12 - On the Ternary Additive Divisor Problem and the Sixth Moment of the Zeta-Function
pp 205-244

13 - A Variant of the Circle Method
pp 245-254

14 - The Resemblance of the Behaviour of the Remainder Terms Eσ(t), Δ1-2σ(x) and R(σ + it)
pp 255-274

15 - A Note on the Number of Divisors of Quadratic Polynomials
pp 275-282

16 - On the Distribution of Integer Points in the Real Locus of an Affine Toric Variety
pp 283-292

17 - An Asymptotic Expansion of the Square of the Riemann Zeta-Function
pp 293-308

18 - The Mean Square of Dedekind Zeta-Functions of Quadratic Number Fields
pp 309-324

19 - Artin's Conjecture and Elliptic Analogues
pp 325-344

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Sieve Methods, Exponential Sums, and their Applications in Number Theory

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