A course in analytic number theory / Marius Overholt.
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...
Saved in:
| Main Author: | |
|---|---|
| Language: | English |
| Published: |
Providence, Rhode Island :
American Mathematical Society,
[2014]
|
| Series: | Graduate studies in mathematics ;
v. 160. |
| Subjects: | |
| Physical Description: | xviii, 371 pages : illustrations ; 26 cm. |
| Format: | Book |
MARC
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| 100 | 1 | |a Overholt, Marius, |d 1957- |e author. |0 http://id.loc.gov/authorities/names/n2014046706 | |
| 245 | 1 | 2 | |a A course in analytic number theory / |c Marius Overholt. |
| 264 | 1 | |a Providence, Rhode Island : |b American Mathematical Society, |c [2014] | |
| 300 | |a xviii, 371 pages : |b illustrations ; |c 26 cm. | ||
| 336 | |a text |b txt |2 rdacontent | ||
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| 490 | 1 | |a Graduate studies in mathematics ; |v volume 160 | |
| 504 | |a Includes bibliographical references and index. | ||
| 505 | 0 | 0 | |a Introduction -- 1: Arithmetic functions -- The method of Chebyshev -- Bertrand's postulate -- Simple estimation techniques -- The Mertens estimates -- Sums over divisors -- the hyperbola method -- Notes -- 2: Topics on arithmetic functions -- The neighborhood method -- The normal order method -- The Mertens functions -- Notes -- 3: Characters and Euler products -- The Euler product formula -- Convergence of Dirichlet series -- Harmonics -- Group representations -- Fourier analysis on finite groups -- Primes in arithmetic porgressions -- Gauss sums and primitive characters -- Ther character group -- Notes -- 4: The circle method -- Diophantine equations -- The major arcs -- The singular series -- Weyl sums -- An asymptotic estimate -- Notes -- 5: The method of contour integrals -- The Perron formula -- Bounds for Dirichlet L-functions -- Notes -- 6: The Prime Number Theorem -- A zero-free region -- A proof of o fht PNT -- Notes -- 7: The Siegel-Walfisz Theorem -- Zero-free regions for L-functions -- An idea of Landau -- The theorem of Siegel -- Teh Borel-Caratheodory lemme -- The PNT for arithmetic progressions -- Notes -- 8: Mainly Analysis -- The Poisson summatino formula -- Theta functions -- The gamma function -- The functional equation of (s) -- The functional equation of L(s,x) -- The Hadamard factorization theorem -- The Phragmen-Lindelof principle -- Notes -- 9: 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 represntaitons -- Artin L-functions -- Notes -- 10: Explicit Formulas -- The von Mangoldt formula -- The primes and RH -- The Guinand-Weil formula -- Notes -- 11: Supplementary Exercises -- Exercises -- Solutions. |
| 520 | |a 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. --Provided by publisher. | ||
| 650 | 0 | |a Number theory. |0 http://id.loc.gov/authorities/subjects/sh85093222 | |
| 650 | 0 | |a Arithmetic functions. |0 http://id.loc.gov/authorities/subjects/sh85007174 | |
| 830 | 0 | |a Graduate studies in mathematics ; |v v. 160. |0 http://id.loc.gov/authorities/names/n92111274 | |
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