Number Theory and Its Applications II

The aim of the book is to give a smooth analytic continuation from basic subjects including linear algebra, group theory, Hilbert space theory, etc. to number theory. With plenty of practical examples and worked-out exercises, and the scope ranging from these basic subjects made applicable to number-theoretic settings to advanced number theory, this book can then be read without tears. It will be of immense help to the reader to acquire basic sound skills in number theory and its applications.

Number theory used to be described as the queen of mathematics, that is, there is no practical use. However, with the development of computers and the security of internet communications, the importance of number theory has been exponentially increasing daily. The raison d'être of the present book in this situation is that it is extremely reader-friendly while keeping the rigor of serious mathematics and in-depth analysis of practical applications to various subjects including control theory and pseudo-random number generation. The use of operators is prevailing rather abundantly in anticipation of applications to electrical engineering, allowing the reader to master these skills without much difficulty. It also delivers a very smooth bridging between elementary subjects including linear algebra and group theory (and algebraic number theory) for the reader to be well-versed in an efficient and effortless way. One of the main features of the book is that it gives several different approaches to the same topic, helping the reader to gain deeper insight and comprehension. Even just browsing through the materials would be beneficial to the reader.

Contents:

• Linear Algebraic Approach to Algebraic Number Theory
• Group-Theoretic Aspects of Algebraic Number Theory
• Arithmetic Functions and Stieltjes Integrals
• Number Theory in the Unit Disc
• Hilbert Space and Number Theory
• Control Systems and Number Theory

Readership: Advanced undergraduate students, graduate students and researchers interested in the use of operators through number-theoretic examples, number-theoretic control systems, applications of number theory, etc.
Key Features:

• Starting from linear algebraic approach to algebraic number theory, we move on to Hilbert space theory which is then used in elucidating automorphic L-functions and control theory and the reader can master the skills without too much difficulty
• In treating summatory functions in number theory, the Stieltjes integration is included which will clarify and unify many techniques used in practice and the elucidation of Lau's theorem on convolution is highlighted. Also given is the Abel–Tauber theorem which applies to an extreme case of the Selberg–Delange type as well as number theory in the unit disc which treats the limiting behavior of modular forms near the real line as that of radial approach to the unit circle. The same aspect is expounded in control theory
• Many examples and worked-out exercises will help the reader to master the fundamental skills
• Since the book gives some different approaches to the same topic and so just browsing through it is beneficial