Analytic Number Theory

Transcription

1 American Mathematical Society Colloquium Publications Volume 53 Analytic Number Theory Henryk Iwaniec Emmanuel Kowalski American Mathematical Society Providence, Rhode Island

2 Contents Preface xi Introduction 1 Chapter 1. Arithmetic Functions Notation and definitions Generating series Dirichlet convolution Examples Arithmetic functions on average Sums of multiplicative functions Distribution of additive functions 28 Chapter 2. Elementary Theory of Prime Numbers The Prime Number Theorem Tchebyshev method Primes in arithmetic progressions Reflections on elementary proofs of the Prime Number Theorem 38 Chapter 3. Characters Introduction Dirichlet characters Primitive characters Gauss sums Real characters The quartic residue symbol The Jacobi-Dirichlet and the Jacobi-Kubota symbols Hecke characters 56 Chapter 4. Summation Formulas Introduction The Euler-Maclaurin formula The Poisson summation formula Summation formulas for the ball Summation formulas for the hyperbola ' Functional equations of Dirichlet L-functions 84 4.A. Appendix: Fourier integrals and series 86 Chapter 5. Classical Analytic Theory of L-functions Definitions and preliminaries 93

3 vi CONTENTS 5.2. Approximations to L-functions Counting zeros of //-functions The zero-free region Explicit formula The prime number theorem The Grand Riemann Hypothesis Simple consequences of GRH The Riemann zeta function and Dirichlet L-functions L-functions of number fields Classical automorphic L-functions General automorphic L-functions Artin L-functions L-functions of varieties A. Appendix: complex analysis 149 Chapter 6. Elementary Sieve Methods Sieve problems Exclusion-inclusion scheme Estimations of V+(z), V~(z) Fundamental Lemma of sieve theory The A 2 -Sieve Estimate for the main term of the A 2 -sieve Estimates for the remainder term in the A 2 -sieve Selected applications of A 2 -sieve 166 Chapter 7. Bilinear Forms and the Large Sieve General principles of estimating double sums Bilinear forms with exponentials Introduction to the large sieve Additive large sieve inequalities Multiplicative large sieve inequality Applications of the large sieve to sieving problems Panorama of the large sieve inequalities Large sieve inequalities for cusp forms Orthogonality of elliptic curves Power moments of L-functions 194 Chapter 8. Exponential Sums Introduction Weyl's method Van der Corput method Discussion of exponent pairs Vinogradov's method 216 Chapter 9. The Dirichlet Polynomials Introduction The integral mean-value estimates The discrete mean-value estimates Large values of Dirichlet polynomials Dirichlet polynomials with characters 238

4 TABLE OF CONTENTS vii 9.6. The reflection method Large values of D(s, \) 246 Chapter 10. Zero Density Estimates Introduction Zero-detecting polynomials Breaking the zero-density conjecture Grand zero-density theorem The gaps between primes 264 Chapter 11. Sums over Finite Fields Introduction Finite fields Exponential sums The Hasse-Davenport relation The zeta function for Kloosterman sums Stepanov's method for hyperelliptic curves Proof of Weil's bound for Kloosterman sums The Riemann Hypothesis for elliptic curves over finite fields Geometry of elliptic curves The local zeta function of elliptic curves Survey of further results: a cohomological primer Comments 313 Chapter 12. Character Sums Introduction Completing methods Complete character sums Short character sums Very short character sums to highly composite modulus Characters to powerful modulus 335 Chapter 13. Sums over Primes General principles A variant of Vinogradov's method Linnik's identity Vaughan's identity Exponential sums over primes Back to the sieve 348 Chapter 14. Holomorphic Modular Forms Quotients of the upper half-plane and modular forms Eisenstein and Poincare series Theta functions Modular forms associated to elliptic curves Hecke L-functions Hecke operators and automorphic L-functions Primitive forms and special basis Twisting modular forms Estimates for the Fourier coefficients of cusp forms 378

5 viii CONTENTS Averages of Fourier coefficients 380 Chapter 15. Spectral Theory of Automorphic Forms Motivation and geometric preliminaries The laplacian on H Automorphic functions and forms The continuous spectrum The discrete spectrum Spectral decomposition and automorphic kernels The Selberg trace formula Hyperbolic lattice point problems Distribution of length of closed geodesies and class numbers 401 Chapter 16. Sums of Kloosterman Sums Introduction Fourier expansion of Poincare series The projection of Poincare series on Maass forms Kuznetsov's formulas Estimates for the Fourier coefficients Estimates for sums of Kloosterman sums 415 Chapter 17. Primes in Arithmetic Progressions Introduction Bilinear forms in arithmetic progressions Proof of the Bombieri-Vinogradov Theorem Proof of the Barban-Davenport-Halberstam Theorem 424 Chapter 18. The Least Prime in an Arithmetic Progression Introduction The log-free zero-density theorem The exceptional zero repulsion Proof of Linnik's Theorem 439 Chapter 19. The Goldbach Problem Introduction Incomplete A-functions A ternary additive problem with A b Proof of Vinogradov's three primes theorem 447 Chapter 20. The Circle Method The partition number Diophantine equations The circle method after Kloosterman Representations by quadratic forms Another decomposition of the delta-symbol 481 Chapter 21. Equidistribution Weyl's criterion Selected equidistribution results Roots of quadratic congruences Linear and bilinear forms in quadratic roots 496

6 TABLE OF CONTENTS ix A Poincare series for quadratic roots Estimation of the Poincare series 501 Chapter 22. Imaginary Quadratic Fields Binary quadratic forms The class group The class group L-functions The class number problems Splitting primes in Q(v /^D) Estimations for derivatives 1^(1,XD) 523 Chapter 23. Effective Bounds for the Class Number Landau's plot of automorphic L-functions A partition of A^)(I) Estimation of S 3 and S Evaluation of Si An asymptotic formula for A^)(i) A lower bound for the class number Concluding notes A The Gross-Zagier L-function vanishes to order Chapter 24. The Critical Zeros of the Riemann Zeta Function A lower bound for N 0 {T) A positive proportion of critical zeros 550 Chapter 25. The Spacing of the Zeros of the Riemann Zeta-Function Introduction The pair correlation of zeros The n-level correlation function for consecutive spacing Low-lying zeros of L-functions 572 Chapter 26. Central Values of L-functions Introduction Principle of the proof of Theorem Formulas for the first and the second moment Optimizing the mollifier Proof of Theorem Bibliography 599 Index 611