Analytic Number Theory: An Introductory Course -- Preface -- Contents -- 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 Dirichlet's 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 Euler's function -- 4.6 Notes -- Chapter 5 An Elementary Proof of the P.N.T. -- 5.1 Selberg's formula -- 5.1.1 Features of Selberg's formula -- 5.2 Transformation of Selberg's formula -- 5.2.1 Calculus for R -- 5.3 Deduction of the P.N.T. -- 5.4 Propositions "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 T(12 * 13) -- 7.10 Notes -- Chapter 8 The Riemann Zeta Function -- 8.1 The functional equation -- 8.1.1 Justification of the interchange of and -- 8.1.2 Symmetric form of the functional equation -- 8.2 O-estimates for zeta -- 8.3 Zeros of zeta -- 8.4 A zerofree region for zeta -- 8.5 An estimate of -- 8.6 Estimation of -- 8.7 The P.N.T. with a remainder term -- 8.8 Estimation of M -- 8.9 The density of zeros in the critical strip -- 8.10 An explicit formula for 1 -- 8.11 Notes -- Chapter 9 Primes in Arithmetic Progressions -- 9.1 Residue characters -- 9.2 Group structure of the coprime residue classes -- 9.3 Existence of enough characters -- 9.4 L functions -- 9.5 Proof of Dirichlet's theorem -- 9.6 P.N.T. for arithmetic progressions -- 9.7 Notes -- Chapter 10 Applications of Characters -- 10.1 Integers generated by primes in residue classes -- 10.2 Sums of squares -- 10.3 A measure of nonprincipality -- 10.4 Quadratic excess -- 10.5 Evaluation of Gaussian sums -- 10.6 Notes -- Chapter 11 Oscillation Theorems -- 11.1 Introduction -- 11.2 Approximate periodicity -- 11.3 The use of Landau's oscillation theorem -- 11.4 A quantitative estimate -- 11.5 The use of many singularities -- 11.5.1 Applications -- 11.6 Sign changes of (x) - li x -- 11.7 The size of M(x)/x -- 11.7.1 Numerical calculations.

11.8 The error term in the divisor problem -- 11.9 Notes -- Chapter 12 Sieves -- 12.1 Introduction -- 12.2 The sieve of Eratosthenes and Legendre -- 12.3 Sieve setup -- 12.4 The Brun-Hooley sieve -- 12.5 The large sieve -- 12.6 An extremal majorant -- 12.7 Proof of Theorem 12.9 -- 12.8 Notes -- Chapter 13 Application of Sieves -- 13.1 A Brun-Hooley estimate of twin primes -- 13.2 The Brun-Titchmarsh inequality -- 13.3 Primes represented by polynomials -- 13.4 A uniform two residue sieve estimate -- 13.5 Twin primes and Goldbach's problem -- 13.6 A heuristic formula for twin primes -- 13.7 Notes -- Appendix A Results from Analysis and Algebra -- A.1 Properties of real functions -- A.1.1 Decomposition -- A.1.2 Riemann-Stieltjes integrals -- A.1.3 Integrators -- A.2 The Euler gamma function -- A.3 Poisson summation formula -- A.4 Basis theorem for finite abelian groups -- Bibliography -- Index of Names and Topics -- Index of Symbols.