Overholt M. A Course in Analytic Number Theory 2014 torrent

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This book is an introduction to analytic number theory suitable for beginning graduate students. It covers everything one expects in a first course in this field, such as growth of arithmetic functions, existence of primes in arithmetic progressions, and the Prime Number Theorem. But it also covers more challenging topics that might be used in a second course, such as the Siegel-Walfisz theorem, functional equations of L-functions, and the explicit formula of von Mangoldt. For students with an interest in Diophantine analysis, there is a chapter on the Circle Method and Waring's Problem. Those with an interest in algebraic number theory may find the chapter on the analytic theory of number fields of interest, with proofs of the Dirichlet unit theorem, the analytic class number formula, the functional equation of the Dedekind zeta function, and the Prime Ideal Theorem.
The exposition is both clear and precise, reflecting careful attention to the needs of the reader. The text includes extensive historical notes, which occur at the ends of the chapters. The exercises range from introductory problems and standard problems in analytic number theory to interesting original problems that will challenge the reader.
The author has made an effort to provide clear explanations for the techniques of analysis used. No background in analysis beyond rigorous calculus and a first course in complex function theory is assumed.
Preface
Acknowledgments
How to use this text
Introduction
Arithmetic Functions
The method of Chebyshev
Bertrand's Postulate
Simple estimation techniques
The Mertens estimates
Sums over divisors
The hyperbola method
Notes
Exercises
Topics on Arithmetic Functions
*The neighborhood method
*The normal order method
*The Mertens function
Notes
Exercises
Characters and Euler Products
The Euler product formula
Convergence of Dirichlet series
Harmonics
Group representations
Fourier analysis on finite groups
Primes in arithmetic progressions
Gauss sums and primitive characters
*The character group
Notes
Exercises
The Circle Method
Diophantine equations
The major arcs
The singular series
Weyl sums
An asymptotic estimate
Notes
Exercises
The Method of Contour Integrals
The Perron formula
Bounds for Dirichlet L-functions
Notes
Exercises
The Prime Number Theorem
A zero-free region
A proof of the PNT
Notes
Exercises
The Siegel-Walfisz Theorem
Zero-free regions for L-functions
An idea of Landau
The theorem of Siegel
The Borel-Caratheodory lemma
The PNT for arithmetic progressions
Notes
Exercises
Mainly Analysis
The Poisson summation formula
Theta functions
The gamma function
The functional equation of ζ(s)
*The functional equation of L(s, χ)
The Hadamard factorization theorem
*The Phragmen-LindelOf principle
Notes
Exercises
Euler Products and Number Fields
The Dedekind zeta function
The analytic class number formula
*Class numbers of quadratic fields
*A discriminant bound
*The Prime Ideal Theorem
*A proof of the Ikehara theorem
Induced representations
Artin L-functions
Notes
Exercises
Explicit Formulas
The von Mangoldt formula
The primes and RH
The Guinand-Weil formula
Notes
Exercises
Supplementary Exercises
Exercises
Solutions
Bibliography
List of Notations
Index