导语

  

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。

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
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