American Mathematical Society Colloquium Publications Volume 53 Analytic Number Theory Henryk Iwaniec Emmanuel Kowalski American Mathematical Society Providence, Rhode Island
Contents Preface xi Introduction 1 Chapter 1. Arithmetic Functions 9 1.1. Notation and definitions 9 1.2. Generating series 10 1.3. Dirichlet convolution 12 1.4. Examples 13 1.5. Arithmetic functions on average 19 1.6. Sums of multiplicative functions 23 1.7. Distribution of additive functions 28 Chapter 2. Elementary Theory of Prime Numbers 31 2.1. The Prime Number Theorem 31 2.2. Tchebyshev method 32 2.3. Primes in arithmetic progressions 34 2.4. Reflections on elementary proofs of the Prime Number Theorem 38 Chapter 3. Characters 43 3.1. Introduction 43 3.2. Dirichlet characters 44 3.3. Primitive characters 45 3.4. Gauss sums 47 3.5. Real characters 49 3.6. The quartic residue symbol 53 3.7. The Jacobi-Dirichlet and the Jacobi-Kubota symbols 55 3.8. Hecke characters 56 Chapter 4. Summation Formulas 65 4.1. Introduction 65 4.2. The Euler-Maclaurin formula 66 4.3. The Poisson summation formula 69 4.4. Summation formulas for the ball 71 4.5. Summation formulas for the hyperbola ' 74 4.6. Functional equations of Dirichlet L-functions 84 4.A. Appendix: Fourier integrals and series 86 Chapter 5. Classical Analytic Theory of L-functions 93 5.1. Definitions and preliminaries 93
vi CONTENTS 5.2. Approximations to L-functions 97 5.3. Counting zeros of //-functions 101 5.4. The zero-free region 105 5.5. Explicit formula 108 5.6. The prime number theorem 110 5.7. The Grand Riemann Hypothesis 113 5.8. Simple consequences of GRH 117 5.9. The Riemann zeta function and Dirichlet L-functions 119 5.10. L-functions of number fields 125 5.11. Classical automorphic L-functions 131 5.12. General automorphic L-functions 136 5.13. Artin L-functions 141 5.14. L-functions of varieties 145 5.A. Appendix: complex analysis 149 Chapter 6. Elementary Sieve Methods 153 6.1. Sieve problems 153 6.2. Exclusion-inclusion scheme 154 6.3. Estimations of V+(z), V~(z) 157 6.4. Fundamental Lemma of sieve theory 158 6.5. The A 2 -Sieve 160 6.6. Estimate for the main term of the A 2 -sieve 164 6.7. Estimates for the remainder term in the A 2 -sieve 165 6.8. Selected applications of A 2 -sieve 166 Chapter 7. Bilinear Forms and the Large Sieve 169 7.1. General principles of estimating double sums 169 7.2. Bilinear forms with exponentials 171 7.3. Introduction to the large sieve 174 7.4. Additive large sieve inequalities 175 7.5. Multiplicative large sieve inequality 179 7.4. Applications of the large sieve to sieving problems 180 7.6. Panorama of the large sieve inequalities 183 7.7. Large sieve inequalities for cusp forms 186 7.8. Orthogonality of elliptic curves 192 7.9. Power moments of L-functions 194 Chapter 8. Exponential Sums 197 8.1. Introduction 197 8.2. Weyl's method 198 8.3. Van der Corput method 204 8.4. Discussion of exponent pairs 213 8.5. Vinogradov's method 216 Chapter 9. The Dirichlet Polynomials 229 9.1. Introduction 229 9.2. The integral mean-value estimates 230 9.3. The discrete mean-value estimates 232 9.4. Large values of Dirichlet polynomials 235 9.5. Dirichlet polynomials with characters 238
TABLE OF CONTENTS vii 9.6. The reflection method 243 9.7. Large values of D(s, \) 246 Chapter 10. Zero Density Estimates 249 10.1. Introduction 249 10.2. Zero-detecting polynomials 250 10.3. Breaking the zero-density conjecture 254 10.4. Grand zero-density theorem 256 10.5. The gaps between primes 264 Chapter 11. Sums over Finite Fields 269 11.1. Introduction 269 11.2. Finite fields 269 11.3. Exponential sums 272 11.4. The Hasse-Davenport relation 274 11.5. The zeta function for Kloosterman sums 278 11.6. Stepanov's method for hyperelliptic curves 281 11.7. Proof of Weil's bound for Kloosterman sums 287 11.8. The Riemann Hypothesis for elliptic curves over finite fields 290 11.9. Geometry of elliptic curves 291 11.10. The local zeta function of elliptic curves 297 11.11. Survey of further results: a cohomological primer 300 11.12. Comments 313 Chapter 12. Character Sums 317 12.1. Introduction 317 12.2. Completing methods 318 12.3. Complete character sums 319 12.4. Short character sums 324 12.5. Very short character sums to highly composite modulus 330 12.6. Characters to powerful modulus 335 Chapter 13. Sums over Primes 337 13.1. General principles 337 13.2. A variant of Vinogradov's method 340 13.3. Linnik's identity 342 13.4. Vaughan's identity 344 13.5. Exponential sums over primes 345 13.6. Back to the sieve 348 Chapter 14. Holomorphic Modular Forms 353 14.1. Quotients of the upper half-plane and modular forms 353 14.2. Eisenstein and Poincare series 357 14.3. Theta functions 361 14.4. Modular forms associated to elliptic curves 363 14.5. Hecke L-functions 368 14.6. Hecke operators and automorphic L-functions 370 14.7. Primitive forms and special basis 372 14.8. Twisting modular forms 376 14.9. Estimates for the Fourier coefficients of cusp forms 378
viii CONTENTS 14.10. Averages of Fourier coefficients 380 Chapter 15. Spectral Theory of Automorphic Forms 383 15.1. Motivation and geometric preliminaries 383 15.2. The laplacian on H 385 15.3. Automorphic functions and forms 386 15.4. The continuous spectrum 387 15.5. The discrete spectrum 389 15.6. Spectral decomposition and automorphic kernels 391 15.7. The Selberg trace formula 393 15.8. Hyperbolic lattice point problems 398 15.9. Distribution of length of closed geodesies and class numbers 401 Chapter 16. Sums of Kloosterman Sums 403 16.1. Introduction 403 16.2. Fourier expansion of Poincare series 404 16.3. The projection of Poincare series on Maass forms 406 16.4. Kuznetsov's formulas 406 16.5. Estimates for the Fourier coefficients 413 16.6. Estimates for sums of Kloosterman sums 415 Chapter 17. Primes in Arithmetic Progressions 419 17.1. Introduction 419 17.2. Bilinear forms in arithmetic progressions 421 17.3. Proof of the Bombieri-Vinogradov Theorem 423 17.4. Proof of the Barban-Davenport-Halberstam Theorem 424 Chapter 18. The Least Prime in an Arithmetic Progression 427 18.1. Introduction 427 18.2. The log-free zero-density theorem 429 18.3. The exceptional zero repulsion 434 18.4. Proof of Linnik's Theorem 439 Chapter 19. The Goldbach Problem 443 19.1. Introduction 443 19.2. Incomplete A-functions 445 19.3. A ternary additive problem with A b 446 19.4. Proof of Vinogradov's three primes theorem 447 Chapter 20. The Circle Method 449 20.1. The partition number 449 20.2. Diophantine equations 456 20.3. The circle method after Kloosterman 467 20.4. Representations by quadratic forms 472 20.5. Another decomposition of the delta-symbol 481 Chapter 21. Equidistribution 487 21.1. Weyl's criterion 487 21.2. Selected equidistribution results 488 21.3. Roots of quadratic congruences 494 21.4. Linear and bilinear forms in quadratic roots 496
TABLE OF CONTENTS ix 21.5. A Poincare series for quadratic roots 498 21.6. Estimation of the Poincare series 501 Chapter 22. Imaginary Quadratic Fields 503 22.1. Binary quadratic forms 503 22.2. The class group 508 22.3. The class group L-functions 511 22.4. The class number problems 517 22.5. Splitting primes in Q(v /^D) 520 22.6. Estimations for derivatives 1^(1,XD) 523 Chapter 23. Effective Bounds for the Class Number 529 23.1. Landau's plot of automorphic L-functions 529 23.2. A partition of A^)(I) 531 23.3. Estimation of S 3 and S 2 533 23.4. Evaluation of Si 534 23.5. An asymptotic formula for A^)(i) 536 23.6. A lower bound for the class number 538 23.7. Concluding notes 540 23.A The Gross-Zagier L-function vanishes to order 3 541 Chapter 24. The Critical Zeros of the Riemann Zeta Function 547 24.1. A lower bound for N 0 {T) 547 24.2. A positive proportion of critical zeros 550 Chapter 25. The Spacing of the Zeros of the Riemann Zeta-Function 563 25.1. Introduction 563 25.2. The pair correlation of zeros 564 25.3. The n-level correlation function for consecutive spacing 570 25.4. Low-lying zeros of L-functions 572 Chapter 26. Central Values of L-functions 577 26.1. Introduction 577 26.2. Principle of the proof of Theorem 26.2 580 26.3. Formulas for the first and the second moment 582 26.4. Optimizing the mollifier 589 26.5. Proof of Theorem 26.2 595 Bibliography 599 Index 611