Contents of Multiplicative Number Theory I

Multiplicative Number Theory I: Classical Theory

 

Contents

 
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
 
12Explicit 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

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