解析数论(影印版) Henryk Iwaniec, Emmanuel Kowal 著 文教 文轩网

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