The aim of this book is to systematize and survey in an easily accessible manner the most important results from some parts of Number Theory, which are connected with many other fields of Mathematics or Science. Each chapter can be viewed as an encyclopedia of the considered field, with many facets and interconnections with vir- tually almost all major topics as Discrete mathematics, Combinatorial theory, Numer- ical analysis, Finite difference calculus, Probability theory; and such classical fields of mathematics as Algebra, Geometry, and Mathematical analysis. Some aspects of Chapter 1 and 3 on Perfect numbers and Euler’s totient, have been considered also in our former volume ”Handbook of Number Theory” (Kluwer Academic Publishers, 1995), in cooperation with the late Professor D. S. Mitrinovic ́ of Belgrade University, as well as Professor B. Crstici, formerly of Timis ̧oara Technical University. However, there were included mainly estimates and inequalities, which are indeed very useful, but many important relations (e.g. congruences) were left out, giving a panoramic view of many other parts of Number Theory.
This volume aims also to complement these issues, and also to bring to the atten- tion of the readers (specialists or not) the hidden beauty of many theories outside a given field of interest.
This book focuses too, as the former volume, on some important arithmetic func- tions of Number Theory and Discrete mathematics, such as Euler’s totient φ(n) and its many generalizations; the sum of divisors function σ(n) with the many old and new issues on Perfect numbers; the Mo ̈bius function, along with its generalizations and extensions, in connection with many applications; the arithmetic functions re- lated to the divisors, consecutive divisors, or the digits of a number. The last chapter shows perhaps most strikingly the cross-fertilization of Number theory with Combi- natorics, Numerical mathematics, or Probability theory.