Preface | page xi | |
List of notation | xiii | |
1 | Dirichlet Series: I | 1 |
1.1 | Generating functions and asymptotics | 1 |
1.2 | Analytic properties of Dirichlet series | 11 |
1.3 | Euler products and the zeta function | 19 |
1.4 | Notes | 31 |
1.5 | References | 33 |
2 | The elementary theory of arithmetic functions | 35 |
2.1 | Mean values | 35 |
2.2 | The prime number estimates of Chebyshev and of Mertens | 46 |
2.3 | Applications to arithmetic functions | 54 |
2.4 | The distribution of Omega(n) - omega(n) | 65 |
2.5 | Notes | 68 |
2.6 | References | 71 |
3 | Principles and first examples of sieve methods | 76 |
3.1 | Initiation | 76 |
3.2 | The Selberg lambda-square method | 82 |
3.3 | Sifting an arithmetic progression | 89 |
3.4 | Twin primes | 91 |
3.5 | Notes | 101 |
3.6 | References | 104 |
4 | Primes in arithmetic progressions: I | 108 |
4.1 | Additive characters | 108 |
4.2 | Dirichlet characters | 115 |
4.3 | Dirichlet L-functions | 120 |
4.4 | Notes | 133 |
4.5 | References | 134 |
5 | Dirichlet series: II | 137 |
5.1 | The inverse Mellin transform | 137 |
5.2 | Summability | 147 |
5.3 | Notes | 162 |
5.4 | References | 164 |
6 | The Prime Number Theorem | 168 |
6.1 | A zero-free region | 168 |
6.2 | Teh Prime Number Theorem | 179 |
6.3 | Notes | 162 |
6.4 | References | 164 |
7 | Applications of the Prime Number Theorem | 199 |
7.1 | Numbers composed of small primes | 199 |
7.2 | Numbers composed of large primes | 215 |
7.3 | Primes in short intervals | 220 |
7.4 | Numbers composed of a prescribed number of primes | 228 |
7.5 | Notes | 239 |
7.6 | References | 241 |
8 | Further discussion of the Prime Number Theorem | 244 |
8.1 | Relations equivalent to the Prime Number Theorem | 244 |
8.2 | An elementary proof of the Prime Number Theorem | 250 |
8.3 | The Wiener-Ikehara Tauberian theorem | 259 |
8.4 | Beurling's generalized prime numbers | 266 |
8.5 | Notes | 276 |
8.6 | References | 279 |
9 | Primitive characters and Gauss sums | 282 |
9.1 | Primitive characters | 282 |
9.2 | Gauss sums | 286 |
9.3 | Quadratic characters | 295 |
9.4 | Incomplete character sums | 306 |
9.5 | Notes | 321 |
9.6 | References | 323 |
10 | Analytic properties of the zeta function and L-functions | 326 |
10.1 | Functional equations and analytic continuation | 326 |
10.2 | Products and sums over zeros | 345 |
10.3 | Notes | 356 |
10.4 | References | 356 |
11 | Primes in arithmetic progression: II | 358 |
11.1 | A zero-free region | 358 |
11.2 | Exceptional zeros | 367 |
11.3 | The Prime Number Theorem for arithmetic progressions | 377 |
11.4 | Applications | 386 |
11.5 | Notes | 391 |
11.6 | References | 393 |
12 | Explicit formulae | 397 |
12.1 | Classical formulae | 397 |
12.2 | Weil's explicit formula | 410 |
12.3 | Notes | 416 |
12.4 | References | 417 |
13 | Conditional estimates | 419 |
13.1 | Estimates for primes | 419 |
13.2 | Estimates for the zeta function | 433 |
13.3 | Notes | 447 |
13.4 | References | 449 |
14 | Zeros | 452 |
14.1 | General distribution of zeros | 452 |
14.2 | Zeros on the critical line | 456 |
14.3 | Notes | 460 |
14.4 | References | 461 |
15 | Oscillations of error terms | 463 |
15.1 | Applications of Landau's theorem | 463 |
15.2 | The error term in the Prime Number Theorem | 475 |
15.3 | Notes | 482 |
15.4 | References | 484 |
APPENDICES |
A | The Riemann-Stieltjes integral | 486 |
A.1 | Notes | 492 |
A.2 | References | 493 |
B | Bernoulli numbers and the Euler-MacLaurin | |
summation formula | 495 | |
B.1 | Notes | 513 |
B.2 | References | 517 |
C | Teh gamma function | 520 |
C.1 | Notes | 531 |
C.2 | References | 533 |
D | Topics in harmonic analysis | 535 |
D.1 | Pointwise convergence of Fourier series | 535 |
D.2 | The Poisson summation formula | 538 |
D.3 | Notes | 542 |
D.4 | References | 542 |
Name index | 544 | |
Subject index | 550 |