Analytic number theory : lectures given at the C.I.M.E. summer school held in Cetraro, Italy, July 11-18, 2002 / J.B. Friedlander ... [and others] ; editors: A. Perelli, C. Viola.

Format:

Book

Contributor:

Friedlander, J. B. (John B.)

Perelli, A. (Alberto)

Viola, C. (Carlo)

Centro internazionale matematico estivo.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1891.

Lecture notes in mathematics, 0075-8434 ; 1891

Language:

English

Subjects (All):

Number theory.

Physical Description:

xi, 212 pages ; 24 cm.

Place of Publication:

Berlin : Springer ; Firenze : Fondazione C.I.M.E., 2006.

Contents:

Producing Prime Numbers via Sieve Methods / John B. Friedlander 1

1 "Classical" sieve methods 2

2 Sieves with cancellation 18

3 Primes of the form X[superscript 2] + Y[superscript 4] 28

4 Asymptotic sieve for primes 38

Counting Rational Points on Algebraic Varieties / D. R. Heath-Brown 51

1 First lecture. A survey of Diophantine equations 51

1.3 The heuristic bounds 53

1.4 Curves 55

1.5 Surfaces 55

1.6 Higher dimensions 57

2 Second lecture. A survey of results 57

2.1 Early approaches 57

2.2 The method of Bombieri and Pila 58

2.3 Projective curves 59

2.4 Surfaces 61

2.5 A general result 64

2.6 Affine problems 64

3 Third lecture. Proof of Theorem 14 65

3.1 Singular points 65

3.2 The Implicit Function Theorem 66

3.3 Vanishing determinants of monomials 68

3.4 Completion of the proof 71

4 Fourth lecture. Rational points on projective surfaces 72

4.1 Theorem 6 - Plane sections 72

4.2 Theorem 6 - Curves of degree 3 or more 73

4.3 Theorem 6 - Quadratic curves 74

4.4 Theorem 8 - Large solutions 74

4.5 Theorem 8 - Inequivalent representations 76

4.6 Theorem 8 - Points on the surface E = 0 77

5 Fifth lecture. Affine varieties 78

5.1 Theorem 15 - The exponent set [epsilon] 78

5.2 Completion of the proof of Theorem 15 79

5.3 Power-free values of polynomials 82

6 Sixth lecture. Sums of powers, and parameterizations 85

6.1 Theorem 13 - Equal sums of two powers 86

6.2 Parameterization by elliptic functions 89

6.3 Sums of three powers 91

Conversations on the Exceptional Character / Henryk Iwaniec 97

2 The exceptional character and its zero 98

3 How was the class number problem solved? 101

4 How and why do the central zeros work? 104

5 What if the GRH holds except for real zeros? 108

6 Subnormal gaps between critical zeros 109

7 Fifty percent is not enough! 112

8 Exceptional primes 114

9 The least prime in an arithmetic progression 117

9.2 The case with an exceptional character 120

9.3 A parity-preserving sieve inequality 123

9.4 Estimation of [psi kappa](x; q, a) 125

9.6 Appendix. Character sums over triple-primes 128

Axiomatic Theory of L-Functions: the Selberg Class / Jerzy Kaczorowski 133

1 Examples of L-functions 134

1.1 Riemann zeta-function and Dirichlet L-functions 134

1.2 Hecke L-functions 136

1.3 Artin L-functions 140

1.4 GL[subscript 2] L-functions 145

1.5 Representation theory and general automorphic L-functions 155

2 The Selberg class: basic facts 159

2.1 Definitions and initial remarks 159

2.2 The simplest converse theorems 163

2.3 Euler product 166

2.4 Factorization 170

2.5 Selberg conjectures 174

3 Functional equation and invariants 177

3.1 Uniqueness of the functional equation 177

3.2 Transformation formulae 178

3.3 Invariants 181

4 Hypergeometric functions 186

4.1 Gauss hypergeometric function 186

4.2 Complete and incomplete Fox hypergeometric functions 187

4.3 The first special case: [mu] = 0 188

4.4 The second special case: [mu] > 0 191

5 Non-linear twists 193

5.1 Meromorphic continuation 193

5.2 Some consequences 196

6 Structure of the Selberg class: d = 1 197

6.1 The case of the extended Selberg class 197

6.2 The case of the Selberg class 200

7 Structure of the Selberg class: 1< d< 2 201

7.1 Basic identity 201

7.2 Fourier transform method 202

7.3 Rankin-Selberg convolution 204

7.4 Non existence of L-functions of degrees 1< d< 5/3 205

7.5 Dulcis in fundo 206.

Notes:

Includes bibliographical references.

ISBN:

3540363637

OCLC:

71747744

Publisher Number:

9783540363637