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'etre 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.

This book emphasizes the role of symmetry and presents as many viewpoints as possible of an important phenomenon - the functional equation of the associated zeta-function. It starts from the basics before warping into the space of new interest; from the ground state to the excited state. For example, the celebrated Gauss quadratic reciprocity law is proved in four independent ways, which are in some way or other dependent on the functional equation. The proofs rest on finite fields, representation theory of nilpotent groups, reciprocity law for the Dedekind sums, and the translation formula for the theta-series, respectively. Likewise, for example, the Euler function is treated in several different places.One of the important principles of learning is to work with the material many times. This book presents many worked-out examples and exercises to enhance the reader's comprehension on the topics covered in an in-depth manner. This is done in a different setting each time such that the reader will always be challenged. For the keen reader, even browsing the text alone, without solving the exercises, will yield some knowledge and enjoyment.