正版 解析数论--2002年在意大利切特拉罗举行的C.I.M.E暑期班演讲

Producing Prime Numbers via Sieve Methods
John B. Friedlander
1 "Classical" sieve methods
2 Sieves with cancellation
3 Primes of the form X2 ~ y4
4 Asymptotic sieve for primes
5 Conclusion
References
Counting Rational Points on Algebraic Varieties
D. R. Heath-Brown
1 First lecture. A survey of Diophantine equations
1.1 Introduction
1.2 Examples
1.3 The heuristic bounds
1.4 Curves
1.5 Surfaces
1.6 Higher dimensions
2 Second lecture. A survey of results
2.1 Early approaches
2.2 The method of Bombieri and Pila
2.3 Projective curves
2.4 Surfaces
2.5 A general result
2.6 Affine problems
3 Third lecture. Proof of Theorem 14
3.1 Singular points
3.2 The Implicit Function Theorem
3.3 Vanishing determinants of monomials
3.4 Completion of the proof
4 Fourth lecture. Rational points on projective surfaces
4.1 Theorem 6 - Plane sections
4.2 Theorem 6 - Curves of degree 3 or more
4.3 Theorem 6 - Quadratic curves
4.4 Theorem 8 - Large solutions
4.5 Theorem 8 - Inequivalent representations
4.6 Theorem 8 - Points on the surface E = 0
5 Fifth lecture. Affine varieties
5.1 Theorem 15 - The exponent set ε
5.2 Completion of the proof of Theorem 15
5.3 Power-free values of polynomials
6 Sixth lecture. Sums of powers, and parameterizations
6.1 Theorem 13 - Equal sums of two powers
6.2 Parameterization by elliptic functions
6.3 Sums of three powers
References
Conversations on the Exceptional Character
Henryk Iwaniec
1 Introduction
2 The exceptional character and its zero
3 How was the class number problem solved?
4 How and why do the central zeros work?
5 What if the GRH holds except for real zeros?
6 Subnormal gaps between critical zeros
7 Fifty percent is not enough!
8 Exceptional primes
9 The least prime in an arithmetic progression
9.1 Introduction
9.2 The case with an exceptional character
9.3 A parity-preserving sieve inequality
9.4 Estimation of ψx(x;q,a)
9.5 Conclusion
9.6 Appendix. Character sums over triple-primes
References
Axiomatic Theory of L-Functions: the Selberg Class
Yerzy Kaczorowski
1 Examples of L-functions
1.1 Riemann zeta-function and Dirichlet L-functions
1.2 Hecke L-functions
1.3 Artin L-functions
1.4 GL2 L-functions
1.5 Representation theory and general automorphic L-functions
2 The Selberg class: basic facts
2.1 Definitions and initial remarks
2.2 The simplest converse theorems
2.3 Euler product
2.4 Factorization
2.5 Selberg conjectures
3 Functional equation and invariants
3.1 Uniqueness of the functional equation
3.2 Transformation formulae
3.3 Invariants
4 Hypergeometric functions
4.1 Gauss hypergeometric function
4.2 Complete and incomplete Fox hypergeometric functions
4.3 The first special case: p = 0
4.4 The second special case: μ > 0
5 Non-linear twists
5.1 Meromorphic continuation
5.2 Some consequences
6 Structure of the Selberg class: d = 1
6.1 The case of the extended Selberg class
6.2 The case of the Selberg class
7 Structure of the Selberg class: 1 < d < 2
7.1 Basic identity
7.2 Fourier transform method
7.3 Rankin-Selberg convolution
7.4 Non existence of L-functions of degrees 1 < d < 5/3
7.5 Dulcis in fundo
References
编辑手记

本书主要介绍分析数论中前沿成果的论文， Heath-Brown的讲义主要介绍了计算丢番图方程的整数解，并阐述了代数几何和数字几何的相关应用；Iwaniec的论文，介绍了西格尔零点理论和L-函数特殊性质的相关推广，并给出了关于算术级数中最小素数的Linnik定理的新证明；Kaczorowski的文章，介绍了Selberg引入的L-函数理论的最新研究成果。