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

(1)

Again

(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

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