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Prime-Detecting Sieves (LMS-33) [Mīkstie vāki]

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  • Formāts: Paperback / softback, 384 pages, height x width: 235x156 mm, 10 b/w illus. 9 tables.
  • Sērija : London Mathematical Society Monographs
  • Izdošanas datums: 26-May-2020
  • Izdevniecība: Princeton University Press
  • ISBN-10: 0691202990
  • ISBN-13: 9780691202990
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Prime-Detecting Sieves (LMS-33)Palielināt
  • Formāts: Paperback / softback, 384 pages, height x width: 235x156 mm, 10 b/w illus. 9 tables.
  • Sērija : London Mathematical Society Monographs
  • Izdošanas datums: 26-May-2020
  • Izdevniecība: Princeton University Press
  • ISBN-10: 0691202990
  • ISBN-13: 9780691202990
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780691202990.html
Keywords:

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 Sieves breaks 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.

Recenzijas

"This book provides a very nice introduction to a very active and important area of research. Several chapters include discussion of the limitations of the given methods; this is an unusual feature but a very useful one to readers. There is also helpful discussion of historical developments of the given methods. This a valuable book, both for researchers and for advanced graduate students in analytic number theory."---S. W Graham, Mathematical Reviews "[ T]his book contains a valuable compendium of methods and results, and it will be of interest to aficionados of prime number theory."---Harold G. Diamond, SIAM Review "The book is written in a very accessible style for a wide spectrum of readers. . . . Besides mathematical ideas, the presentation also contains many important historical comments, which make the book useful for a general mathematical audience trying to orient themselves in the evolution of the main techniques applied in sieve methods." * EMS Newsletter *

Preface xi
Notation xiii
Chapter 1 Introduction
1(24)
1.1 The Beginning
1(3)
1.2 The Sieve of Eratosthenes
4(2)
1.3 The Sieve of Eratosthenes-Legendre
6(2)
1.4 The Prime Number Theorem and Its Consequences
8(10)
1.5 Brun, Selberg, and Rosser-Iwaniec
18(2)
1.6 Eratosthenes-Legendre-Vinogradov
20(5)
Chapter 2 The Vaughan Identity
25(22)
2.1 Introduction
25(3)
2.2 An Exponential Sum over Primes
28(1)
2.3 The Distribution of αp Modulo 1
29(4)
2.4 The Bombieri-Vinogradov Theorem
33(5)
2.5 Linnik's and Heath-Brown's Identities
38(4)
2.6 Further Thoughts on Vaughan's Identity
42(5)
Chapter 3 The Alternative Sieve
47(18)
3.1 Introduction
47(2)
3.2 Cosmetic Surgery
49(1)
3.3 The Fundamental Theorem
50(4)
3.4 Application to the Distribution of {αp}
54(2)
3.5 A Lower-Bound Sieve
56(4)
3.6 A Change of Notation
60(2)
3.7 The Piatetski-Shapiro PNT
62(1)
3.8 Historical Note
63(2)
Chapter 4 The Rosser-Iwaniec Sieve
65(18)
4.1 Introduction
65(2)
4.2 A Fundamental Lemma
67(5)
4.3 A Heuristic Argument
72(1)
4.4 Proof of the Lower-Bound Sieve
73(6)
4.5 Developments of the Rosser-Iwaniec Sieve
79(4)
Chapter 5 Developing the Alternative Sieve
83(20)
5.1 Introduction
83(1)
5.2 New Forms of the Fundamental Theorem
83(3)
5.3 Reversing Roles
86(5)
5.4 A New Idea
91(1)
5.5 Higher-Dimensional Versions
92(1)
5.6 Greatest Prime Factors
93(10)
Chapter 6 An Upper-Bound Sieve
103(16)
6.1 The Method Described
103(2)
6.2 A Device by Chebychev
105(2)
6.3 The Arithmetical Information
107(3)
6.4 Applying the Rosser-Iwaniec Sieve
110(2)
6.5 An Asymptotic Formula
112(1)
6.6 The Alternative Sieve Applied
112(3)
6.7 Upper-Bounds: Region by Region
115(3)
6.8 Why a Previous Idea Fails
118(1)
Chapter 7 Primes in Short Intervals
119(38)
7.1 The Zero-Density Approach
119(2)
7.2 Preliminary Results
121(7)
7.3 The 7/12 Result
128(5)
7.4 Shorter Intervals
133(2)
7.5 Application of Watt's Theorem
135(4)
7.6 Sieve Asymptotic Formulae
139(4)
7.7 The Two-Dimensional Sieve Revisited
143(4)
7.8 Further Asymptotic Formulae
147(3)
7.9 The Final Decomposition
150(5)
7.10 Where to Now?
155(2)
Chapter 8 The Brun-Titchmarsh Theorem on Average
157(32)
8.1 Introduction
157(2)
8.2 The Arithmetical Information
159(6)
8.3 The Alternative Sieve Applied
165(7)
8.4 The Alternative Sieve for τ ≤ α1 ≤ 3/7, θ ≤ 11/21
172(2)
8.5 The Alternative Sieve in Two Dimensions
174(4)
8.6 The Alternative Sieve in Three Dimensions
178(4)
8.7 An Upper Bound for Large θ
182(1)
8.8 Completion of Proof
183(6)
Chapter 9 Primes in Almost All Intervals
189(12)
9.1 Introduction
189(2)
9.2 The Arithmetical Information
191(4)
9.3 The Alternative Sieve Applied
195(3)
9.4 The Final Decomposition
198(1)
9.5 An Upper-Bound Result
199(1)
9.6 Other Measures of Gaps Between Primes
200(1)
Chapter 10 Combination with the Vector Sieve
201(30)
10.1 Introduction
201(1)
10.2 Goldbach Numbers in Short Intervals
202(3)
10.3 Proof of Theorem 10.2
205(6)
10.4 Dirichlet Polynomials
211(7)
10.5 Sieving the Interval B1
218(9)
10.6 Sieving the Intervals B2
227(2)
10.7 Further Applications
229(2)
Chapter 11 Generalizing to Algebraic Number Fields
231(34)
11.1 Introduction
231(1)
11.2 Gaussian Primes in Sectors
232(1)
11.3 Notation and Outline of the Method
233(4)
11.4 The Arithmetical Information
237(3)
11.5 Asymptotic Formulae for Problem 1
240(4)
11.6 The Final Decomposition for Problem 1
244(3)
11.7 Prime Ideals in Small Regions
247(1)
11.8 First Steps
248(7)
11.9 Estimates for Dirichlet Polynomials
255(3)
11.10 Asymptotic Formulae for Problem 2
258(2)
11.11 The Final Decomposition for Problem 2
260(5)
Chapter 12 Variations on Gaussian Primes
265(38)
12.1 Introduction
265(1)
12.2 Outline of the Fouvry-Iwaniec Method
266(2)
12.3 Some Preliminary Results
268(5)
12.4 Fouvry-Iwaniec Type I Information
273(3)
12.5 Reducing the Bilinear Form Problem
276(3)
12.6 Catching the Cancellation Introduced by μ
279(5)
12.7 The Main Term for Theorem 12.1
284(1)
12.8 The Friedlander-Iwaniec Outline for a2 + b4
285(2)
12.9 The Friedlander-Iwaniec Asymptotic Sieve
287(9)
12.10 Sketch of the Crucial Result
296(5)
12.11 And Now?
301(2)
Chapter 13 Primes of the Form x3 + 2y3
303(32)
13.1 Introduction
303(1)
13.2 Outline of the Proof
304(8)
13.3 Preliminary Results
312(1)
13.4 The Type I Estimates
313(3)
13.5 The Fundamental Lemma Result
316(1)
13.6 Proof of Lemma 13.6
317(8)
13.7 Proof of Lemma 13.7
325(1)
13.8 The Type II Information Established
326(9)
Chapter 14 Epilogue
335(2)
14.1 A Summary
335(1)
14.2 A Challenge with Which to Close
336(1)
Appendix
337(12)
A.1 Perron's formula
337(2)
A.2 Buchstab's Function ω(u)
339(4)
A.3 Large-Sieve Inequalities
343(3)
A.4 The Mean Value Theorem for Dirichlet Polynomials
346(1)
A.5 Smooth Functions
347(2)
Bibliography 349(12)
Index 361
Glyn Harman is professor of pure mathematics at the University of London, Royal Holloway. He is the author of Metric Number Theory, the coeditor of Sieve Methods, Exponential Sums, and their Applications in Number Theory, and the corecipient of the Hardy-Ramanujan award for his work on primes in short intervals.