Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

On Quasi Multiplicative Function

Azizul Hoque*, Himashree Kalita*

Department of Mathematics, Gauhati University, Guwahati, IndiaDepartment of Mathematics, Gauhati University, Guwahati, India

Abstract: In this paper we introduce two new Arithmetic functions, that is, Quasi-Multiplicative (QM) and omega functions. The Omega function is based on Euler’s Phi function and is used to find the sum of coprime integers. Euler’s Phi function, Dedekind’s psi function, the sigma function and -function play significant role in this work.

Keywords: arithmetic function, quasi-multiplicative function, omega function

Received March 21, 2015; Revised April 07, 2015; Accepted June 02, 2015

Citation: Azizul Hoque, Himashree Kalita, On Quasi Multiplicative Function. Turkish Journal of Analysis and Number Theory, 3 (2). 68-69. January 2015. doi:10.12691/tjant-3-2-6

1. Introduction

Recall that an Arithmetic function [2,6] is Multiplicative if for each pair of coprime integers and , it is satisfied . Missana [5] established some significant results on multiplicative functions. Dehaye [3] constructed some algebraic structures using arithmetic functions. Some classical examples of multiplicative functions that have important meaning in Number theory are Euler’s Phi function, Dedekind’s psi function, the sigma function and -function. Recently Hoque and Kalita [4] studied perfect numbers and their generalizations using these multiplicative functions. For any Positive integer having the factorization , where are distinct prime numbers and are integers, these functions admit the following multiplicative representations:

In this paper, we introduce the notions of Omega function and quasi-multiplicative function. The Omega function is based on Euler’s Phi function and is used to find the sum of coprime integers. We establish some important results on these two functions via Euler’s Phi function, Dedekind’s psi function, the sigma function and -function.

2. Main Results

Definition 2.1: An Arithmetic function is Quasi-Multiplicative (QM) if for each pair of coprime integers and there exists a positive integer such that . The positive is defined as the multiplicative index of .

It is clear that a quasi-multiplicative function with index one is Multiplicative.

Definition 2.2: For any positive integer , we define the Omega function, as the sum of the positive integers less than and relatively prime to .

By Theorem 7.7 of [2], we see that .

Proposition 2.1: If is a multiplicative function and is a positive integer then is quasi-multiplicative function with index .

Proof: Let and be any two positive and coprime integers. Then

This proves the result.

Theorem 2.2: The function is quasi multiplicative with index 2.

Proof: By definition,

Since is multiplicative, the function defined by is also multiplicative.

Thus by Proposition 2.1, is quasi multiplicative with index 2.

Proposition 2.3: For any prime number ,

Proof: We have

[picture] | (1)

Again

[picture] | (2)

From equation (1) and equation (2), the result follows.

Lemma 2.4 [1]: For every natural number ,

Proposition 2.5: For every natural number ,

Theorem 2.6: There are infinitely many positive integers and such that

(i)

(ii)

(iii)

Proof. (i) Let us suppose, for any positive integer . Then

Now,

Also, for .

Thus for .

Moreover, for .

(ii) Let for any positive integer and . Then

Now, .

Also .

(iii) Let for any positive integer . Then

Thus .

Theorem 2.7: There are infinitely many positive integers and such that

(i)

(ii)

(iii)

Proof. (i) Let for any positive integers and . Then

Now, .

And, for .

Thus we have .

(ii) Let for any positive integer and . Then

Now, for

And, .

Thus we have .

(iv) Let for any positive integer . Then

Now, and .

Acknowledgements

First author acknowledges UGC for Junior Research Fellowship (No.GU/UGC/VI(3)/JRF/ 2012/2985).

References

[1]   | Atanassov, K., Notes on ,ψ and σ-functions. Part 6, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 1, 22-24.
[2]   | Burton, D. M., Elementary Number Theory, 6th edi., Tata McGraw-Hill Pub.Com. Ltd, New Delhi.
[3]   | Dehaye, P. O., On the structure of the group of multiplicative Arithmetic functions, Bull. Belg. Math. Soc., 9, 15-21, 2002.
[4]   | Hoque, A. and Kalita, H., Generalised perfect numbers connected with Arithmetic functions, Math. Sci. Lett., 3(3), 249-253, 2014.
[5]   | Missana, M. V., Some results on multiplicative functions, Notes on Number Theory and Discrete Mathematics, 16 (4), 22-24, 2010.
[6]   | Sivaramakrishnan, R., Classical Theory of Arithmetic Function, New York, Dekker, 1989.