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Excursions in Multiplicative Number Theory Birkhäuser Advanced Texts Basler Lehrbücher by Olivier Ramaré 9783030731694 3030731693

Name: Excursions in Multiplicative Number Theory Birkhäuser Advanced Texts Basler Lehrbücher by Olivier Ramaré 9783030731694 3030731693
SKU: EB-51992466
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Excursions in Multiplicative Number Theory Birkhäuser Advanced Texts Basler Lehrbücher Olivier Ramaré Digital Instant Download

Author(s): Olivier Ramaré

Edition: 1st ed. 2022

File Details: PDF, 3.53 MB

Year: 2022

Language: english

SKU: EB-51992466 Category: Mathematics - Number Theory Tags: 3030731693, 9783030731694, Multiplicative, Number Theory, Olivier Ramaré

Description

Excursions in Multiplicative Number Theory Birkhäuser Advanced Texts Basler Lehrbücher Olivier Ramaré – Ebook PDF Instant Download/Delivery, ISBN: 9783030731694, 3030731693

Product details:

ISBN 10:3030731693
ISBN 13:9783030731694
Author: Olivier Ramaré

Excursions in Multiplicative Number Theory

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.

Table contents:

Part I. Approach: Multiplicativity
1. Arithmetic Convolution
2. A Calculus on Arithmetical Functions
3. Analytical Dirichlet Series
4. Growth of Arithmetical Functions
5. An “Algebraical” Multiplicative Function
6. Möbius Inversions
7. Handling a Smooth Factor
Part II. The Convolution Walk
8. The Convolution Method
9. Euler Products and Euler Sums
10. Some Practice
11. Dirichlet’s Hyperbola Principle
12. The Mertens Estimates
Part III. The Levin-Fainleib Walk
13. The Levin–Faĭnleĭb Theorem and Analogues
14. Variations on a Theme of Chebyshev
15. Primes in Arithmetical Progressions
16. Computing a Famous Constant
17. Euler Products with Primes in Progressions
18. The Chinese Remainder Theorem and Multiplicativity
19. The Riemann Zeta Function
Part IV. The Mellin Walk
20. The Mellin Transform
21. Proof of Theorem
22. Roughing up: Removing a Smoothening
23. Proving the Prime Number Theorem
24. The Selberg Formula
Part V. Higher Ground: Applications/Extensions
25. Rankin’s Trick and Brun’s Sieve
26. Three Arithmetical Exponential Sums
27. Convolution Method and Non-Positive Functions
28. The Large Sieve Inequality
29. Montgomery’s Sieve