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 spe case: p = 0 4.4 The second spe 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
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