Excursions in multiplicative number theory / Olivier Ramaré ; with contributions by Pieter Moree and Alisa Sedunova.
Ramaré, Olivier, author.
2022
QA241 .R36 2022
Available Online

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Title Excursions in multiplicative number theory / Olivier Ramaré ; with contributions by Pieter Moree and Alisa Sedunova.
Author Ramaré, Olivier, author.
ISBN 9783030731694 (electronic bk.)
3030731693 (electronic bk.)
9783030731687
3030731685
DOI https://doi.org/10.1007/978-3-030-73169-4
Published Cham : Birkhäuser, [2022]
Copyright ©2022
Language English
Description 1 online resource : illustrations (chiefly color).
Item Number 10.1007/978-3-030-73169-4 doi
Call Number QA241 .R36 2022
Dewey Decimal Classification 512.7
Summary 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 as the Selberg formula, Exponential sums with arithmetical coefficients, 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.
Bibliography, etc. Note Includes bibliographical references and indexes.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (SpringerLink, viewed March 9, 2022).
Series Birkhäuser advanced texts. 2296-4894
Available in Other Form Print version: 9783030731687
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Approach: Multiplicativity
Arithmetic Convolution
A Calculus on Arithmetical Functions
Analytical Dirichlet Series
Growth of Arithmetical Functions
An "Algebraical" Multiplicative Function
Möbius Inversions
The Convolution Walk
Handling a Smooth Factor
The Convolution Method
Euler Products and Euler Sums
Some Practice
The Hyperbola Principle
The Levin-Faĭnleĭb Walk
The Mertens Estimates
The Levin-Fanleib Theorem
Variations on a Theme of Chebyshev
Primes in progressions
A famous constant
Euler Products with Primes in AP
Chinese Remainder and Multiplicativity
The Mellin Walk
The Riemann zeta-function
The Mellin Transform
Proof Theorem Á
Roughing up: Removing a Smoothening
Proving the Prime Number Theorem
Higher Ground: Applications / Extensions
The Selberg Formula
Rankin's Trick and Brun's Sieve
Three Arithmetical Exponential Sums
Convolution method / Möbius function
The Large Sieve Inequality
Montgomery's Sieve.

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