Analytic number theory an introductory course - Bateman, P. T. (World Scientific, 2004)
Includes bibliographical references (pages 353-354) and indexes
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
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