Analytic Number Theory: An Introductory Course (Monographs In Number Theory Book 1)

This valuable book focuses on a collection of powerful methods of analysis that yield deep number-theoretical estimates. Particular attention is given to counting functions of prime numbers and multiplicative arithmetic functions. Both real variable ("elementary") and complex variable ("analytic") methods are employed. The reader is assumed to have knowledge of elementary number theory (abstract algebra will also do) and real and complex analysis. Specialized analytic techniques, including transform and Tauberian methods, are developed as needed.Comments and corrigenda for the book are found at (http://www.math.uiuc.edu/) www.math.uiuc.edu/ diamond/. Cover Title - Analytic number theory 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 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(12x13) 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