Prime Detecting Sieves

by Harman, Glyn

• ISBN13: 9780691124377
• ISBN10: 069112437X
• Format: Hardcover
• Copyright: 2007-07-16
• Publisher: Princeton Univ Pr

Summary

This book seeks to describe the rapid development in recent decades of sieve methods able to detect prime numbers. The subject began with Eratosthenes in antiquity, took on new shape with Legendre's form of the sieve, was substantially reworked by Ivan M. Vinogradov and Yuri V. Linnik, but came into its own with Robert C. Vaughan and important contributions from others, notably Roger Heath-Brown and Henryk Iwaniec.Prime-Detecting Sievesbreaks new ground by bringing together several different types of problems that have been tackled with modern sieve methods and by discussing the ideas common to each, in particular the use of Type I and Type II information. No other book has undertaken such a systematic treatment of prime-detecting sieves. Among the many topics Glyn Harman covers are primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes, and the recent work of John Friedlander and Iwaniec on primes that are a sum of a square and a fourth power, and Heath-Brown's work on primes represented as a cube plus twice a cube. This book contains much that is accessible to beginning graduate students, yet also provides insights that will benefit established researchers.

Author Biography

Glyn Harman is professor of pure mathematics at the University of London, Royal Holloway.

Table of Contents

Preface	p. xi
Notation	p. xiii
Introduction	p. 1
The Beginning	p. 1
The Sieve of Eratosthenes	p. 4
The Sieve of Eratosthenes-Legendre	p. 6
The Prime Number Theorem and Its Consequences	p. 8
Brun, Selberg, and Rosser-Iwaniec	p. 18
Eratosthenes-Legendre-Vinogradov	p. 20
The Vaughan Identity	p. 25
Introduction	p. 25
An Exponential Sum over Primes	p. 28
The Distribution of [alpha]p Modulo 1	p. 29
The Bombieri-Vinogradov Theorem	p. 33
Linnik's and Heath-Brown's Identities	p. 38
Further Thoughts on Vaughan's Identity	p. 42
The Alternative Sieve	p. 47
Introduction	p. 47
Cosmetic Surgery	p. 49
The Fundamental Theorem	p. 50
Application to the Distribution of {[alpha]p}	p. 54
A Lower-Bound Sieve	p. 56
A Change of Notation	p. 60
The Piatetski-Shapiro PNT	p. 62
Historical Note	p. 63
The Rosser-Iwaniec Sieve	p. 65
Introduction	p. 65
A Fundamental Lemma	p. 67
A Heuristic Argument	p. 72
Proof of the Lower-Bound Sieve	p. 73
Developments of the Rosser-Iwaniec Sieve	p. 79
Developing the Alternative Sieve	p. 83
Introduction	p. 83
New Forms of the Fundamental Theorem	p. 83
Reversing Roles	p. 86
A New Idea	p. 91
Higher-Dimensional Versions	p. 92
Greatest Prime Factors	p. 93
An Upper-Bound Sieve	p. 103
The Method Described	p. 103
A Device by Chebychev	p. 105
The Arithmetical Information	p. 107
Applying the Rosser-Iwaniec Sieve	p. 110
An Asymptotic Formula	p. 112
The Alternative Sieve Applied	p. 112
Upper-Bounds: Region by Region	p. 115
Why a Previous Idea Fails	p. 118
Primes in Short Intervals	p. 119
The Zero-Density Approach	p. 119
Preliminary Results	p. 121
The 7/12 Result	p. 128
Shorter Intervals	p. 133
Application of Watt's Theorem	p. 135
Sieve Asymptotic Formulae	p. 139
The Two-Dimensional Sieve Revisited	p. 143
Further Asymptotic Formulae	p. 147
The Final Decomposition	p. 150
Where to Now?	p. 155
The Brun-Titchmarsh Theorem on Average	p. 157
Introduction	p. 157
The Arithmetical Information	p. 159
The Alternative Sieve Applied	p. 165
The Alternative Sieve for [tau less than or equal alpha]1 [less than or equal] 3/7, [theta] [less than or equal] 11/21	p. 172
The Alternative Sieve in Two Dimensions	p. 174
The Alternative Sieve in Three Dimensions	p. 178
An Upper Bound for Large [theta]	p. 182
Completion of Proof	p. 183
Primes in Almost All Intervals	p. 189
Introduction	p. 189
The Arithmetical Information	p. 191
The Alternative Sieve Applied	p. 195
The Final Decomposition	p. 198
An Upper-Bound Result	p. 199
Other Measures of Gaps Between Primes	p. 200
Combination with the Vector Sieve	p. 201
Introduction	p. 201
Goldbach Numbers in Short Intervals	p. 202
Proof of Theorem	p. 205
Dirichlet Polynomials	p. 211
Sieving the Interval B[subscript 1]	p. 218
Sieving the Interval B[subscript 2]	p. 227
Further Applications	p. 229
Generalizing to Algebraic Number Fields	p. 231
Introduction	p. 231
Gaussian Primes in Sectors	p. 232
Notation and Outline of the Method	p. 233
The Arithmetical Information	p. 237
Asymptotic Formulae for Problem 1	p. 240
The Final Decomposition for Problem 1	p. 244
Prime Ideals in Small Regions	p. 247
First Steps	p. 248
Estimates for Dirichlet Polynomials	p. 255
Asymptotic Formulae for Problem 2	p. 258
The Final Decomposition for Problem 2	p. 260
Variations on Gaussian Primes	p. 265
Introduction	p. 265
Outline of the Fouvry-Iwaniec Method	p. 266
Some Preliminary Results	p. 268
Fouvry-Iwaniec Type I Information	p. 273
Reducing the Bilinear Form Problem	p. 276
Catching the Cancellation Introduced by [Mu]	p. 279
The Main Term for Theorem 12.1	p. 284
The Friedlander-Iwaniec Outline for a[superscript a] + b[superscript 4]	p. 285
The Friedlander-Iwaniec Asymptotic Sieve	p. 287
Sketch of the Crucial Result	p. 296
And Now?	p. 301
Primes of the Form x[superscript 3] + 2y[superscript 3]	p. 303
Introduction	p. 303
Outline of the Proof	p. 304
Preliminary Results	p. 312
The Type I Estimates	p. 313
The Fundamental Lemma Result	p. 316
Proof of Lemma 13.6	p. 317
Proof of Lemma 13.7	p. 325
The Type II Information Established	p. 326
Epilogue	p. 335
A Summary	p. 335
A Challenge with Which to Close	p. 336
Appendix	p. 337
Perron's formula	p. 337
Buchstab's Function [omega](u)	p. 339
Large-Sieve Inequalities	p. 343
The Mean Value Theorem for Dirichlet Polynomials	p. 346
Smooth Functions	p. 347
Bibliography	p. 349
Index	p. 361
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