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Author Bateman, Paul Trevier
Title Analytic Number Theory : An Introductory Course
Imprint Singapore : World Scientific Publishing Company, 2004
©2004
book jacket
Descript 1 online resource (375 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series Monographs In Number Theory ; v.1
Monographs In Number Theory
Note Intro -- 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
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
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Bateman, Paul Trevier Analytic Number Theory: An Introductory Course Singapore : World Scientific Publishing Company,c2004 9789812389381
Subject Number theory.;Mathematical analysis
Electronic books
Alt Author Diamond, Harold G
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