## 解析数论--2002年在意大利切特拉罗举行的C.I.M.E暑期班演讲集(英文版)/国外优秀数学著作原版系列

• 定价： ￥68
• ISBN：9787560386683
• 开 本：16开 平装
• 作者：(加)J.B.弗里德兰...
• 立即节省：
• 2020-06-01 第1版
• 2020-06-01 第1次印刷

### 内容提要

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。

### 目录

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