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Equivalents of the Riemann Hypothesis 2 Hardback Volume Set [Multiple-component retail product]

(University of Waikato, New Zealand)
  • Formatas: Multiple-component retail product, aukštis x plotis x storis: 241x160x56 mm, weight: 1540 g, Worked examples or Exercises; 26 Tables, black and white; 33 Halftones, black and white; 51 Line drawings, black and white, Contains 2 hardbacks
  • Serija: Encyclopedia of Mathematics and its Applications
  • Išleidimo metai: 02-Nov-2017
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108290787
  • ISBN-13: 9781108290784
Kitos knygos pagal šią temą:
Equivalents of the Riemann Hypothesis 2 Hardback Volume SetDidesnis paveikslėlis
  • Formatas: Multiple-component retail product, aukštis x plotis x storis: 241x160x56 mm, weight: 1540 g, Worked examples or Exercises; 26 Tables, black and white; 33 Halftones, black and white; 51 Line drawings, black and white, Contains 2 hardbacks
  • Serija: Encyclopedia of Mathematics and its Applications
  • Išleidimo metai: 02-Nov-2017
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108290787
  • ISBN-13: 9781108290784
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781108290784.html
The Riemann hypothesis (RH) is perhaps the most important outstanding problem in mathematics. This two-volume text presents the main known equivalents to RH using analytic and computational methods. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between different results. It also includes extensive tables, supplementary computational tools, and open problems suitable for research. Accompanying software is free to download. These books will interest mathematicians who wish to update their knowledge, graduate and senior undergraduate students seeking accessible research problems in number theory, and others who want to explore and extend results computationally. Each volume can be read independently. Volume 1 presents classical and modern arithmetic equivalents to RH, with some analytic methods. Volume 2 covers equivalences with a strong analytic orientation, supported by an extensive set of appendices containing fully developed proofs.

This two-volume work covers the main known equivalents to the Riemann hypothesis. Classical and modern arithmetic statements equivalent to the Riemann hypothesis are elucidated, proved and applied using analytic and computational methods. Each volume can be read independently.

Recenzijos

'This two volume catalogue of many of the various equivalents of the Riemann Hypothesis by Kevin Broughan is a valuable addition to the literature all in all these two volumes are a must have for anyone interested in the Riemann Hypothesis.' Steven Decke, MAA Reviews 'Throughout the book careful proofs are given for all the results discussed, introducing an impressive range of mathematical tools. Indeed, the main achievement of the work is the way in which it demonstrates how all these diverse subject areas can be brought to bear on the Riemann hypothesis. The exposition is accessible to strong undergraduates, but even specialists will find material here to interest them.' D. R. Heath-Brown, Mathematical Reviews 'This two volume catalogue of many of the various equivalents of the Riemann Hypothesis by Kevin Broughan is a valuable addition to the literature all in all these two volumes are a must have for anyone interested in the Riemann Hypothesis.' Steven Decke, MAA Reviews 'The two volumes are a very valuable resource and a fascinating read about a most intriguing problem.' R.S. MacKay, London Mathematical Society Newsletter 'All in all these books serve as a good introduction to a wide range of mathematics related to the Riemann Hypothesis and make for a valuable contribution to the literature. They are truly encyclopedic and I am sure will entice many a reader to consult some literature quoted and who knows, eventually make an own contribution to the area.' Pieter Moree, Nieuw Archief voor Wiskunde

Daugiau informacijos

This two-volume work presents the main known equivalents to the Riemann hypothesis. Each volume can be read independently.
Equivalents of the Riemann Hypothesis-Volume One
Contents for Volume Two
x
List of Illustrations
xiv
List of Tables
xvi
Preface for Volume One
xvii
List of Acknowledgements
xxi
1 Introduction
1(14)
1.1
Chapter Summary
1(1)
1.2 Early History
1(7)
1.3 Volume One Summary
8(4)
1.4 Notation
12(1)
1.5 Background Reading
13(1)
1.6 Unsolved Problems
14(1)
2 The Riemann Zeta Function
15(25)
2.1 Introduction
15(1)
2.2 Basic Properties
16(5)
2.3 Zero-Free Regions
21(4)
2.4 Landau's Zero-Free Region
25(4)
2.5 Zero-Free Regions Summary
29(1)
2.6 The Product Over Zeta Zeros
30(9)
2.7 Unsolved Problems
39(1)
3 Estimates
40(28)
3.1 Introduction
40(1)
3.2 Constructing Tables of Bounds for psi(x)
41(10)
3.3 Exact Verification Using Computation
51(3)
3.4 Estimates for theta(x)
54(11)
3.5 More Estimates
65(2)
3.6 Unsolved Problems
67(1)
4 Classical Equivalences
68(26)
4.1 Introduction
68(1)
4.2 The Prime Number Theorem and Its RH Equivalences
69(12)
4.3 Oscillation Theorems
81(7)
4.4 Errors in Arithmetic Sums
88(5)
4.5 Unsolved Problems
93(1)
5 Euler's Totient Function
94(50)
5.1 Introduction
94(4)
5.2 Estimates for Euler's Function phi(n)
98(12)
5.3 Preliminary Results With RH True
110(13)
5.4 Further Results With RH True
123(7)
5.5 Preliminary Results With RH False
130(5)
5.6 Nicolas' First Theorem
135(2)
5.7 Nicolas' Second Theorem
137(5)
5.8 Unsolved Problems
142(2)
6 A Variety of Abundant Numbers
144(21)
6.1 Introduction
144(3)
6.2 Superabundant Numbers
147(6)
6.3 Colossally Abundant Numbers
153(8)
6.4 Estimates for x2(epsilon)
161(2)
6.5 Unsolved Problems
163(2)
7 Robin's Theorem
165(35)
7.1 Introduction
165(4)
7.2 Ramanujan's Theorem Assuming RH
169(5)
7.3 Preliminary Lemmas With RH True
174(6)
7.4 Bounding Phipless than or equal to x(1-p-2) From Above With RH True
180(4)
7.5 Bounding loglogN From Below With RH True
184(2)
7.6 Proof of Robin's Theorem With RH True
186(2)
7.7 An Unconditional Bound for sigma(n)/n
188(2)
7.8 Bounding loglogN From Above Without RH
190(1)
7.9 A Lower Bound for sigma(n)/n With RH False
191(2)
7.10 Lagarias' Formulation of Robin's Criterion
193(3)
7.11 Unconditional Results for Lagarias' Formulation
196(1)
7.12 Unitary Divisor Sums
197(1)
7.13 Unsolved Problems
198(2)
8 Numbers That Do Not Satisfy Robin's Inequality
200(18)
8.1 Introduction
200(2)
8.2 Hardy-Ramanujan Numbers
202(6)
8.3 Integers Not Divisible by the Fifth Power of Any Prime
208(3)
8.4 Integers Not Divisible by the Seventh Power of Any Prime
211(3)
8.5 Integers Not Divisible by the 11th Power of Any Prime
214(3)
8.6 Unsolved Problems
217(1)
9 Left, Right and Extremely Abundant Numbers
218(18)
9.1 Introduction
218(2)
9.2 Gronwall's Theorem
220(3)
9.3 Further Preliminary Results
223(2)
9.4 Riemann Hypothesis Equivalences
225(7)
9.5 Comparing Colossally and Left Abundant Numbers
232(3)
9.6 Extremely Abundant Numbers
235(1)
9.7 Unsolved Problems
235(1)
10 Other Equivalents to the Riemann Hypothesis
236(51)
10.1 Introduction
236(3)
10.2 Shapiro's Criterion
239(2)
10.3 Farey Fractions
241(6)
10.4 Redheffer Matrix
247(3)
10.5 Divisibility Graph
250(2)
10.6 Dirichlet Eta Function
252(1)
10.7 The Derivative of zeta(s)
253(3)
10.8 A Zeta-Related Inequality
256(3)
10.9 The Real Part of the Logarithmic Derivative of xi(s)
259(12)
10.10 The Order of Elements of the Symmetric Group
271(11)
10.11 Hilbert-Polya Conjecture
282(3)
10.12 Epilogue
285(1)
10.13 Unsolved Problems
286(1)
Appendix A: Tables
287(7)
A.1 Extremely Abundant Numbers
287(1)
A.2 Small Numbers Not Satisfying Robin's Inequality
288(1)
A.3 Superabundant Numbers
289(1)
A.4 Colossally Abundant Numbers
290(1)
A.5 Primes to Make Colossally Abundant Numbers
291(1)
A.6 Small Numbers Satisfying Nicolas' Reversed Inequality
292(1)
A.7 Heights of Integers
293(1)
A.8 Maximum Order of an Element of the Symmetric Group
293(1)
Appendix B: RHpack Mini-Manual
294(19)
B.1 Introduction
294(2)
B.2 RHpack Functions
296(17)
References
313(8)
Index
321
Equivalents of the Riemann Hypothesis-Volume Two
Contents for Volume One
xi
List of Illustrations
xiv
List of Tables
xvi
Preface for Volume Two
xvii
List of Acknowledgements
xxi
1 Introduction
1(7)
1.1 Why This Study?
1(1)
1.2 Summary of Volume Two
2(5)
1.3 How to Read This Book
7(1)
2 Series Equivalents
8(15)
2.1 Introduction
8(2)
2.2 The Riesz Function
10(4)
2.3 Additional Properties of the Riesz Function
14(1)
2.4 The Series of Hardy and Littlewood
15(1)
2.5 A General Theorem for a Class of Entire Functions
16(6)
2.6 Further Work
22(1)
3 Banach and Hilbert Space Methods
23(14)
3.1 Introduction
23(2)
3.2 Preliminary Definitions and Results
25(4)
3.3 Beurling's Theorem
29(6)
3.4 Recent Developments
35(2)
4 The Riemann Xi Function
37(25)
4.1 Introduction
37(3)
4.2 Preliminary Results
40(9)
4.3 Monotonicity of |xi(s)|
49(2)
4.4 Positive Even Derivatives
51(3)
4.5 Li's Equivalence
54(5)
4.6 More Recent Results
59(3)
5 The De Bruijn-Newman Constant
62(31)
5.1 Introduction
62(4)
5.2 Preliminary Definitions and Results
66(3)
5.3 A Region for Xilambda(z) With Only Real Zeros
69(8)
5.4 The Existence of Lambda
77(1)
5.5 Improved Lower Bounds for Lambda
77(15)
5.5.1 Lehmer's Phenomenon
78(3)
5.5.2 The Differential Equation Satisfied by H(t,z)
81(6)
5.5.3 Finding a Lower Bound for Ac Using Lehmer Pairs
87(5)
5.6 Further Work
92(1)
6 Orthogonal Polynomials
93(24)
6.1 Introduction
93(1)
6.2 Definitions
94(2)
6.3 Orthogonal Polynomial Properties
96(3)
6.4 Moments
99(5)
6.5 Quasi-Analytic Functions
104(2)
6.6 Carleman's Inequality
106(7)
6.7 Riemann Zeta Function Application
113(3)
6.8 Recent Work
116(1)
7 Cyclotomic Polynomials
117(10)
7.1 Introduction
117(1)
7.2 Definitions
118(1)
7.3 Preliminary Results
119(5)
7.4 Riemann Hypothesis Equivalences
124(2)
7.5 Further Work
126(1)
8 Integral Equations
127(23)
8.1 Introduction
127(2)
8.2 Preliminary Results
129(4)
8.3 The Method of Sekatskii, Beltraminelli and Merlini
133(6)
8.4 Salem's Equation
139(3)
8.5 Levinson's Equivalence
142(8)
9 Weil's Explicit Formula, Inequality and Conjectures
150(43)
9.1 Introduction
150(2)
9.2 Definitions
152(1)
9.3 Preliminary Results
152(2)
9.4 Weil's Explicit Formula
154(5)
9.5 Weil's Inequality
159(7)
9.6 Bombieri's Variational Approach to RH
166(7)
9.7 Introduction to the Weil Conjectures
173(1)
9.8 History of the Weil Conjectures
174(2)
9.9 Finite Fields
176(2)
9.10 The Weil Conjectures for Varieties
178(1)
9.11 Elliptic Curves
178(4)
9.12 Weil Conjectures for Elliptic Curves-Preliminary Results
182(4)
9.13 Proof of the Weil Conjectures for Elliptic Curves
186(2)
9.14 General Curves Over Fq and Applications
188(2)
9.15 Return to the Explicit Formula
190(2)
9.16 Weil's Commentary on his 1952 and 1972 Papers
192(1)
10 Discrete Measures
193(28)
10.1 Introduction
193(1)
10.2 Definitions
194(1)
10.3 Preliminary Results
195(2)
10.4 A Mellin-Style Transform
197(3)
10.5 Verjovsky's Theorems
200(6)
10.6 Historical Development of Non-Euclidean Geometry
206(2)
10.7 The Hyperbolic Upper Half Plane H
208(1)
10.8 The Groups PSL(2,R) and PSL(2,Z)
209(2)
10.9 Eisenstein Series
211(5)
10.10 Zagier's Horocycle Equivalence
216(3)
10.11 Additional Results
219(2)
11 Hermitian Forms
221(53)
11.1 Introduction
221(2)
11.2 Definitions
223(3)
11.3 Distributions
226(2)
11.4 Positive Definite
228(3)
11.5 The Restriction to C(a) for All a > 0
231(5)
11.6 Properties of K(a) and K(a)
236(6)
11.7 Matrix Elements
242(5)
11.8 An Explicit Example With a = log square root of 2
247(11)
11.9 Lemmas for Yoshida's Main Theorem
258(2)
11.10 Hermitian Forms Lemma
260(9)
11.11 Yoshida's Main Theorem
269(1)
11.12 The Restriction to K(a) for All a > 0
270(4)
12 Dirichlet L-Functions
274(58)
12.1 Introduction
274(3)
12.2 Definitions
277(6)
12.3 Properties of L(s,chi)
283(1)
12.4 The Non-Vanishing of L(1,chi)
284(4)
12.5 Zero-Free Regions and Siegel Zeros
288(7)
12.6 Preliminary Results for Titchmarsh's Criterion
295(1)
12.7 Titchmarsh's GRH Equivalence
296(2)
12.8 Preliminary Results for Gallagher's Theorem
298(4)
12.9 Gallagher's Theorems
302(5)
12.10 Applications of Gallagher's Theorems
307(4)
12.11 The Bombieri-Vinogradov Theorem
311(12)
12.12 Applications of Bombieri-Vinogradov's Theorem
323(3)
12.13 Generalizations and Developments for Bombieri-Vinogradov
326(1)
12.14 Conjectures
327(5)
13 Smooth Numbers
332(27)
13.1 Introduction
332(3)
13.2 The Dickman Function
335(11)
13.3 Preliminary Lemmas for Hildebrand's Equivalence
346(3)
13.4 Riemann Hypothesis Equivalence
349(8)
13.5 Further Work
357(2)
14 Epilogue
359(2)
Appendix A: Convergence of Series
361(2)
Appendix B: Complex Function Theory
363(14)
Appendix C: The Riemann-Stieltjes Integral
377(4)
Appendix D: The Lebesgue Integral on Real Numbers
381(7)
Appendix E: The Fourier Transform
388(17)
Appendix F: The Laplace Transform
405(4)
Appendix G: The Mellin Transform
409(9)
Appendix H: The Gamma Function
418(7)
Appendix I: The Riemann Zeta Function
425(17)
Appendix J: Banach and Hilbert Spaces
442(9)
Appendix K: Miscellaneous Background Results
451(8)
Appendix L: GRHpack Mini-Manual
459(14)
L.1 Introduction
459(2)
L.1.1 Installation
459(1)
L.1.2 About This Mini-Manual
460(1)
L.2 GRHpack Functions
461(12)
References
473(12)
Index
485
Kevin Broughan is Emeritus Professor in the Department of Mathematics and Statistics at the University of Waikato, New Zealand. In these two volumes he has used a unique combination of mathematical knowledge and skills. Following the publication of his Columbia University thesis, he worked on problems in topology before undertaking work on symbolic computation, leading to development of the software system SENAC. This led to a symbolic-numeric dynamical systems study of the zeta function, giving new insights into its behaviour, and was accompanied by publication of the software GL(n)pack as part of D. Goldfeld's book, Automorphic Forms and L-Functions for the Group GL(n,R). Professor Broughan has published widely on problems in prime number theory. His other achievements include co-establishing the New Zealand Mathematical Society, the School of Computing and Mathematical Sciences at the University of Waikato, and the basis for New Zealand's connection to the internet.