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Analytic number theory

Author: Henryk Iwaniec; Emmanuel Kowalski
Publisher: Providence, R.I. : American Mathematical Society, ©2004.
Series: Colloquium publications (American Mathematical Society), v. 53.
Edition/Format:   Print book : EnglishView all editions and formats
Database:WorldCat
Summary:
"The book is written with graduate students in mind, and the authors tried to balance between clarity, completeness, and generality. The exercises in each section serve a dual purpose, with some intended to improve the reader's understanding of the subject and others providing additional information. Formal prerequisites for the major part of the book do not go beyond calculus, complex analysis, integration, and  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Henryk Iwaniec; Emmanuel Kowalski
ISBN: 0821836331 9780821836330
OCLC Number: 54806043
Description: ix, 615 p. ; 26 cm.
Contents: Chapter 1. Arithmetic Functions 9 --
1.1. Notation and definitions 9 --
1.2. Generating series 10 --
1.3. Dirichlet convolution 12 --
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 --
3.2. 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 --
4.2. 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 --
5.2. Approximations to L-functions 97 --
5.3. Counting zeros of L-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[superscript +](z), V[superscript -](z) 157 --
6.4. Fundamental Lemma of sieve theory 158 --
6.5. The [Lambda superscript 2]-Sieve 160 --
6.6. Estimate for the main term of the [Lambda superscript 2]-sieve 164 --
6.7. Estimates for the remainder term in the [Lambda superscript 2]-sieve 165 --
6.8. Selected applications of [Lambda superscript 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 --
8.2. 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 --
9.2. 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 --
9.6. The reflection method 243 --
9.7. Large values of D(s, x) 246 --
Chapter 10. Zero Density Estimates 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 --
11.2. 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 --
Chapter 12. Character Sums 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 --
13.2. 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 --
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 geodesics and class numbers 401 --
Chapter 16. Sums of Kloosterman Sums 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 --
17.2. 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 --
18.2. 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 --
19.2. Incomplete [Lambda]-functions 445 --
19.3. A ternary additive problem with [Lambda superscript 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 --
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([radical]D) 520 --
22.6. Estimations for derivatives L[superscript (k)] (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 [Lambda superscript (g)] (1/2) 531 --
23.3. Estimation of S[subscript 3] and S[subscript 2] 533 --
23.4. Evaluation of S[subscript 1] 534 --
23.5. An asymptotic formula for [Lambda superscript (g)] (1/2) 536 --
23.6. A lower bound for the class number 538 --
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[subscript 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 --
25.2. 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 --
26.2. Principle of the proof of Theorem 26.2 580 --
26.3. Formulas for the first and the second moment 582 --
26.4. Optimizing the mollifer 589 --
26.5. Proof of Theorem 26.2 595.
Series Title: Colloquium publications (American Mathematical Society), v. 53.
Responsibility: Henryk Iwaniec, Emmanuel Kowalski.

Abstract:

Analytic Number Theory distinguishes itself by the variety of tools it uses to establish results. This work aims to show the scope of the theory, both in classical and modern directions, and to  Read more...

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