Authors:
- Offers a unique exploration of analytic number theory that focuses on proving explicit bounds in cases suited to versatile tools
- Emphasizes a methodological approach to the material with several different pathways to proceed
- Promotes an active learning style with nearly 300 exercises appearing throughout
Part of the book series: Birkhäuser Advanced Texts Basler Lehrbücher (BAT)
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Table of contents (29 chapters)
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Front Matter
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The Convolution Walk
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Front Matter
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The Levin-Fainleib Walk
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Front Matter
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About this book
This textbook offers a unique exploration of analytic number theory that is focused on explicit and realistic numerical bounds. By giving precise proofs in simplified settings, the author strategically builds practical tools and insights for exploring the behavior of arithmetical functions. An active learning style is encouraged across nearly three hundred exercises, making this an indispensable resource for both students and instructors.
Designed to allow readers several different pathways to progress from basic notions to active areas of research, the book begins with a study of arithmetic functions and notions of arithmetical interest. From here, several guided “walks” invite readers to continue, offering explorations along three broad themes: the convolution method, the Levin–Faĭnleĭb theorem, and the Mellin transform. Having followed any one of the walks, readers will arrive at “higher ground”, where they will find opportunities for extensions and applications, such asthe Selberg formula, Brun’s sieve, and the Large Sieve Inequality. Methodology is emphasized throughout, with frequent opportunities to explore numerically using computer algebra packages Pari/GP and Sage.
Excursions in Multiplicative Number Theory is ideal for graduate students and upper-level undergraduate students who are familiar with the fundamentals of analytic number theory. It will also appeal to researchers in mathematics and engineering interested in experimental techniques in this active area.
Keywords
- Analytic number theory
- Prime number theory
- Euler products
- Multiplicative functions number theory
- Multiplicative arithmetic functions
- Divisor functions
- Convolution of divisor functions
- Riemann zeta function
- Dirichlet series
- Unitary convolution
- Möbius inversion
- Legendre symbol
- Convolutions method number theory
- Levin Fainleib theorem
- Mellin transform
- Large sieve number theory
- Exponential sums with arithmetical coefficients
- Selberg formula
Reviews
“What a wonderful book! If you’re a number theorist with a slight aversion to the more technical parts of analytic number theory, then this book is the proper remedy.” (Franz Lemmermeyer, zbMATH 1496.11003, 2022)
“It does touch on a wealth of topics and techniques. … The book is easy to read. … the book is thoroughly footnoted, including references to the original papers and modern expositions;” (Allen Stenger, MAA Reviews, May 9, 2022)
Authors and Affiliations
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Institut de Mathématiques de Marseille, CNRS / Aix-Marseille University, Marseille, Bouches du Rhône, France
Olivier Ramaré
About the author
Olivier Ramaré is a Research Director at Aix Marseille Université in Marseille, France. He is a prolific researcher with a focus on sieve theory, prime numbers, the Möbius function, L-series, and more.
Bibliographic Information
Book Title: Excursions in Multiplicative Number Theory
Authors: Olivier Ramaré
Series Title: Birkhäuser Advanced Texts Basler Lehrbücher
DOI: https://doi.org/10.1007/978-3-030-73169-4
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
Hardcover ISBN: 978-3-030-73168-7Published: 04 March 2022
Softcover ISBN: 978-3-030-73171-7Published: 05 March 2023
eBook ISBN: 978-3-030-73169-4Published: 03 March 2022
Series ISSN: 1019-6242
Series E-ISSN: 2296-4894
Edition Number: 1
Number of Pages: XXII, 338
Number of Illustrations: 2 b/w illustrations, 13 illustrations in colour
Topics: Number Theory