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Analytic number theory : an introductory course

Author: P T Bateman; Harold G Diamond
Publisher: New Jersey : World Scientific, ©2004.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Database:WorldCat
Summary:
This valuable book focuses on a collection of powerful methods ofanalysis that yield deep number-theoretical estimates. Particularattention is given to counting functions of prime numbers andmultiplicative arithmetic functions. Both real variable ("elementary")and complex variable ("analytic") methods are employed.
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Bateman, P.T.
Analytic number theory.
New Jersey : World Scientific, ©2004
(OCoLC)57420268
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: P T Bateman; Harold G Diamond
ISBN: 9789812389381 9812389385 9789812560803 9812560807 9812562273 9789812562272
OCLC Number: 61482715
Description: 1 online resource (xiii, 360 pages) : illustrations
Contents: Cover --
Contents --
Preface --
Chapter 1 Introduction --
1.1 Three problems --
1.2 Asymmetric distribution of quadratic residues --
1.3 The prime number theorem --
1.4 Density of squarefree integers --
1.5 The Riemann zeta function --
1.6 Notes --
Chapter 2 Calculus of Arithmetic Functions --
2.1 Arithmetic functions and convolution --
2.2 Inverses --
2.3 Convergence --
2.4 Exponential mapping --
2.4.1 The 1 function as an exponential --
2.4.2 Powers and roots --
2.5 Multiplicative functions --
2.6 Notes --
Chapter 3 Summatory Functions --
3.1 Generalities --
3.2 Estimate of Q(x) 6x/2 --
3.3 Riemann-Stieltjes integrals --
3.4 Riemann-Stieltjes integrators --
3.4.1 Convolution of integrators --
3.4.2 Generalization of results on arithmetic functions --
3.5 Stability --
3.6 Dirichlets hyperbola method --
3.7 Notes --
Chapter 4 The Distribution of Prime Numbers --
4.1 General remarks --
4.2 The Chebyshev function --
4.3 Mertens estimates --
4.4 Convergent sums over primes --
4.5 A lower estimate for Eulers function --
4.6 Notes --
Chapter 5 An Elementary Proof of the P.N.T. --
5.1 Selbergs formula --
5.1.1 Features of Selbergs formula --
5.2 Transformation of Selbergs formula --
5.2.1 Calculus for R --
5.3 Deduction of the P.N.T. --
5.4 Propositions 8220;equivalent to the P.N.T. --
5.5 Some consequences of the P.N.T. --
5.6 Notes --
Chapter 6 Dirichlet Series and Mellin Transforms --
6.1 The use of transforms --
6.2 Euler products --
6.3 Convergence --
6.3.1 Abscissa of convergence --
6.3.2 Abscissa of absolute convergence --
6.4 Uniform convergence --
6.5 Analyticity --
6.5.1 Analytic continuation --
6.5.2 Continuation of zeta --
6.5.3 Example of analyticity on = --
6.6 Uniqueness --
6.6.1 Identifying an arithmetic function --
6.7 Operational calculus --
6.8 Landau's oscillation theorem --
6.9 Notes --
Chapter 7 Inversion Formulas --
7.1 The use of inversion formulas --
7.2 The Wiener-Ikehara theorem --
7.2.1 Example. Counting product representations --
7.2.2 An O-estimate --
7.3 A Wiener-Ikehara proof of the P.N.T. --
7.4 A generalization of the Wiener-Ikehara theorem --
7.5 The Perron formula --
7.6 Proof of the Perron formula --
7.7 Contour deformation in the Perron formula --
7.7.1 The Fourier series of the sawtooth function --
7.7.2 Bounded and uniform convergence --
7.8 A "smoothed" Perron formula --
7.9 Example. Estimation of [sigma]T(1₂ * 1₃) --
7.10 Notes --
Chapter 8 The Riemann Zeta Function --
Chapter 9 Primes in Arithmetic Progressions --
Chapter 10 Applications of characters --
Chapter 11 Oscillation theorems --
Chapter 12 Sieves --
Chapter 13 Application of Sieves --
Appendix A. Results from Analysis and Algebra.
Responsibility: Paul T. Bateman, Harold G. Diamond.

Abstract:

This valuable book focuses on a collection of powerful methods ofanalysis that yield deep number-theoretical estimates. Particularattention is given to counting functions of prime numbers andmultiplicative arithmetic functions. Both real variable ("elementary")and complex variable ("analytic") methods are employed.

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