Analytic number theory : an introductory course /

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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 ("a...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Bateman, P. T.
Otros Autores: Diamond, Harold G., 1940-
Formato: Electrónico eBook
Idioma:Inglés
Publicado: New Jersey : World Scientific, ©2004.
Temas:
Number theory.
Mathematical analysis.
Théorie des nombres.
Analyse mathématique.
MATHEMATICS > Number Theory.
Mathematical analysis
Number theory
Nombres, Théorie des.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.

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