Arithmetic Functions

This monograph is devoted to arithmetic functions, an area of number theory. Arithmetic functions are very important in many parts of theoretical and applied sciences, and many mathematicians have devoted great interest in this field. This book contains original results by the authors related to Euler’s totient function, the distinct or total number of prime divisors of a number, Dedekind’s arithmetic function, the prime counting function, the core function, and many other classical functions. One of the interesting features of this book is the introduction and study of certain new arithmetic functions that have been considered by the authors separately or together, and their importance is shown in many connections with the classical arithmetic functions or in their applications to other problems. These functions include extension factor, irrational factor, converse factor, restrictive factor, functions analogous to operation logarithm, factorial and derivative, etc. Different equalities and inequalities related to these functions proposed by the two authors are proved. Properties of perfect numbers and related numbers are discussed. A solution is provided for Mullin’s problem. Generalizations of some known problems are proposed by the authors, e.g., modification of Sivaramakrishnan-Venkataraman’s inequality, Atanassov’s generalization of a theorem by József Sándor and Florian Luca.