Analytic number theory : Introductory course

Includes Index

Includes Bibliography (p. 353-354).

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

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.