Multiplicative Number Theory I: Classical Theory,9780521849036

Multiplicative Number Theory I: Classical Theory

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ISBN13: 9780521849036
ISBN10: 0521849039
Format: Hardcover
Pub. Date: 2006-12-11
Publisher(s): Cambridge University Press
List Price: $149.10

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Summary

Prime numbers are the multiplicative building blocks of natural numbers. Understanding their overall influence and especially their distribution gives rise to central questions in mathematics and physics. In particular, their finer distribution is closely connected with the Riemann hypothesis, the most important unsolved problem in the mathematical world. This book comprehensively covers all the topics met in first courses on multiplicative number theory and the distribution of prime numbers. The text is based on courses taught successfully over many years at the University of Michigan, Imperial College, London and Pennsylvania State University.

Author Biography

Robert Vaughan is a Professor of Mathematics at Pennsylvania State University.

Table of Contents

Preface 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 Ω(n) — ω(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-squared 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 The Prime Number Theorem
179
6.3 Notes
192
6.4 References
195
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 progressions: 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 the 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 The 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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