Textbook in PDF format 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