美国数学会经典影印系列解析数论(影印版)

Preface Introduction Chapter 1. Arithmetic Functions 1.1. Notation and definitions 1.2. Generating series 1.3. Dirichlet convolution 1.4. Examples 1.5. Arithmetic functions on average 1.6. Sums of multiplicative functions 1.7. Distribution of additive functions Chapter 2. Elementary Theory of Prime Numbers 2.1. The Prime Number Theorem 2.2. Tchebyshev method 2.3. Primes in arithmetic progressions 2.4. Reflections on elementary proofs of the Prime Number Theorem Chapter 3. Characters 3.1. Introduction 3.2. Dirichlet characters 3.3. Primitive characters 3.4. Gauss sums 3.5. Real characters 3.6. The quartic residue symbol 3.7. The Jacobi-Dirichlet and the Jacobi-Kubota symbols 3.8. Hecke characters Chapter 4. Summation Formulas $4.1. Introduction 4.2. The Euler-Maclaurin formula 4.3. The Poisson summation formula 4.4. Summation formulas for the ball 4.5. Summation formulas for the hyperbola 4.6. Functional equations of Dirichlet L-functions 4.A. Appendix: Fourier integrals and series Chapter 5. Classical Analytic Theory of L-functions 5.1. Definitions and preliminaries 5.2. Approximations to L-functions 5.3. Counting zeros of L-functions 5.4. The zero-free region 5.5. Explicit formula 5.6. The prime number theorem 5.7. The Grand Riemann Hypothesis 5.8. Simple consequences of GRH 5.9. The Riemann zeta function and Dirichlet L-functions 5.10. L-functions of number fields 5.11. Classical automorphic L-functions 5.12. General automorphic L-functions 5.13. Artin L-functions 5.14. L-functions of varieties 5.A. Appendix: complex analysis Chapter 6. Elementary Sieve Methods 6.1. Sieve problems 6.2. Exclusion-inclusion scheme 6.3. Estimations of V+(z), V-(z) 6.4. Fundamental Lemma of sieve theory 6.5. The A2-Sieve 6.6. Estimate for the main term of the A2-sieve 6.7. Estimates for the remainder term in the A2-sieve 6.8. Selected applications of A2-sieve Chapter 7. Bilinear Forms and the Large Sieve 7.1. General principles of estimating double sums 7.2. Bilinear forms with exponentials 7.3. Introduction to the large sieve 7.4. Additive large sieve inequalities 7.5. Multiplicative large sieve inequality 7.4. Applications of the large sieve to sieving problems 7.6. Panorama of the large sieve inequalities 7.7. Large sieve inequalities for cusp forms 7.8. Orthogonality of elliptic curves 7.9. Power moments of L-functions Chapter 8. Exponential Sums 8.1. Introduction 8.2. Weyl's method 8.3. Van der Corput method 8.4. Discussion of exponent pairs 8.5. Vinogradov's method Chapter 9. The Dirichlet Polynomials 9.1. Introduction 9.2. The integral mean-value estimates 9.3. The discrete mean-value estimates 9.4. Large values of Dirichlet polynomials 9.5. Dirichlet polynomials with characters 9.6. The reflection method 9.7. Large values of D(s, X) Chapter 10. Zero Density Estimates 10.1. Introduction 10.2. Zero-detecting polynomials 10.3. Breaking the zero-density conjecture 10.4. Grand zero-density theorem 10.5. The gaps between primes Chapter 11. Sums over Finite Fields 11.1. Introduction 11.2. Finite fields 11.3. Exponential sums 11.4. The Hasse-Davenport relation 11.5. The zeta function for Kloosterman sums 11.6. Stepanov's method for hyperelliptic curves 11.7. Proof of Weil's bound for Kloosterman sums 11.8. The Riemann Hypothesis for elliptic curves over finite fields 11.9. Geometry of elliptic curves 11.10. The local zeta function of elliptic curves 11.11. Survey of further results: a cohomological primer 11.12. Comments Chapter 12. Character Sums 12.1. Introduction 12.2. Completing methods 12.3. Complete character sums 12.4. Short character sums 12.5. Very short character sums to highly composite modulus 12.6. Characters to powerful modulus Chapter 13. Sums over Primes 13.1. General principles 13.2. A variant of Vinogradov's method 13.3. Linnik's identity 13.4. Vaughan's identity 13.5. Exponential sums over primes 13.6. Back to the sieve Chapter 14. Holomorphic Modular Forms 14.1. Quotients of the upper half-plane and modular forms 14.2. Eisenstein and Poincar series 14.3. Theta functions 14.4. Modular forms associated to elliptic curves 14.5. Hecke L-functions 14.6. Hecke operators and automorphic L-functions 14.7, Primitive forms and special basis 14.8. Twisting modular forms 14.9. Estimates for the Fourier coefficients of cusp forms 14.10. Averages of Fourier coefficients Chapter 15. Spectral Theory of Automorphic Forms 15.1. Motivation and geometric preliminaries 15.2. The laplacian on IH[ 15.3. Automorphic functions and forms 15.4. The continuous spectrum 15.5. The discrete spectrum 15.6. Spectral decomposition and automorphic kernels 15.7. The Selberg trace formula 15.8. Hyperbolic lattice point problems 15.9. Distribution of length of closed geodesics and class numbers Chapter 16. Sums of Kloosterman Sums 16.1. Introduction 16.2. Fourier expansion of Poincar@ series 16.3. The projection of Poincar@ series on Maass forms 16.4. Kuznetsov's formulas 16.5. Estimates for the Fourier coefficients 16.6. Estimates for sums of Kloosterman sums Chapter 17. Primes in Arithmetic Progressions 17.1. Introduction 17.2. Bilinear forms in arithmetic progressions 17.3. Proof of the Bombieri-Vinogradov Theorem 17.4. Proof of the Barban-Davenport-Halberstam Theorem Chapter 18. The Least Prime in an Arithmetic Progression 18.1. Introduction 18.2. The log-free zero-density theorem 18.3. The exceptional zero repulsion 18.4. Proof of Linnik's Theorem Chapter 19. The Goldbach Problem 19.1. Introduction 19.2. Incomplete A-functions 19.3. A ternary additive problem with 19.4. Proof of Vinogradov's three primes theorem Chapter 20. The Circle Method 20.1. The partition number 20.2. Diophantine equations 20.3. The circle method after Kloosterman 20.4. Representations by quadratic forms 20.5. Another decomposition of the delta-symbol Chapter 21. Equidistribution 21.1. Weyl's criterion 21.2. Selected equidistribution results 21.3. Roots of quadratic congruences 21.4. Linear and bilinear forms in quadratic roots 21.5. A Poincar series for quadratic roots 21.6. Estimation of the Poincar series Chapter 22. Imaginary Quadratic Fields 22.1. Binary quadratic forms 22.2. The class group 22.3. The class group L-functions 22.4. The class number problems 22.5. Splitting primes in □(数理化公式) 22.6. Estimations for derivatives □(数理化公式) Chapter 23. Effective Bounds for the Class Number 23.1. Landau's plot of automorphic L-functions 23.2. h partition of□(数理化公式) 23.3. Estimation of S3 and S2 23.4. Evaluation of S1 23.5. An asymptotic formula for □(数理化公式) 23.6. A lower bound for the class number 23.7. Concluding notes 23.A The Gross-Zagier L-function vanishes to order 3 Chapter 24. The Critical Zeros of the Riemann Zeta Function 24.1. A lower bound for No(T) 24.2. A positive proportion of critical zeros Chapter 25. The Spacing of the Zeros of the Riemann Zeta-Function 25.1. Introduction 25.2. The pair correlation of zeros 25.3. The n-level correlation function for consecutive spacing 25.4. Low-lying zeros of L-functions Chapter 26. Central Values of L-functions 26.1. Introduction 26.2. Principle of the proof of Theorem 26.2 26.3. Formulas for the first and the second moment 26.4. Optimizing the mollifier 26.5. Proof of Theorem 26.2 Bibliography Index