Arithmetic Functions
- Length: 241 pages
- Edition: 1
- Language: English
- Publisher: Nova Science Pub Inc
- Publication Date: 2021-05-14
- ISBN-10: 1536194751
- ISBN-13: 9781536194753
- Sales Rank: #0 (See Top 100 Books)
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.
ARITHMETIC FUNCTIONS ARITHMETIC FUNCTIONS Contents Glossary of Symbols Preface Chapter 1On Standard ArithmeticFunctions j,y and s 1.1. A. Mullin’s Inequality and Its Modification 1.2. Inequalities Related to j, y and s-Functions 1.3. A Modification of Sivaramakrishnan-Venkataraman’s Inequality 1.4. On the Composition of Some Arithmetic Functions 1.5. On the Equation j(n)+d(n) = n and RelatedInequalities Chapter 2Perfect and Related Numbers 2.1. On (m,n)-Super-Perfect Numbers 2.2. On a Modification of Perfect Numbers 2.3. On Multiplicatively Perfect Numbers 2.4. Other Modifications of the Concept of PerfectNumber 2.5. A New Point of View on Perfect and Other SimilarNumbers 2.6. On Bi-Unitary Harmonic Numbers 2.7. On Modified Hyperperfect Numbers 2.8. On Balanced Numbers Chapter 3On Modifications andExtensions of the ArithmeticFunctions j,y and s 3.1. On an Arithmetic Function, Related to Operation“Logarithm” 3.2. Irrational Factor: Definition, Propertiesand Problems 3.3. Converse Factor: Definition, Propertiesand Problems 3.4. Restrictive Factor: Definition, Propertiesand Problems 3.5. On an Arithmetic Function, Related to Operation“Derivative” 3.6. On an Arithmetic Function Related to Function s 3.7. Extension Factor: Definition, Propertiesand Problems 3.8. Extensions of Restrictive and Extension Factors 3.8.1. First Round of Generalizations 3.8.2. Second Round of Generalizations 3.8.3. Additive Analogues Chapter 4Arithmetic Functions of OtherTypes 4.1. A Digital Arithmetic Function 4.2. On an Inequality of Klamkin, Its ArithmeticApplications,Modifications and Extension 4.3. Some Representations Related to Arithmetic Function“Factorial” 4.4. Some Representations Concerning the Product ofDivisors of n 4.5. A Note on Certain Euler–Mascheroni TypeSequences 4.6. On Two Conjectures by K. Kashihara on PrimeNumbers 4.7. A Generalization of J´ozsef S´andor and FlorianLuca’s Theorem References About the Authors Index Blank Page
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