Mladen Vassilev-Missana and Peter Vassilev

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 17, 2011, Number 2, Page 18—30

**Download full paper: PDF, 266 Kb**

## Details

### Authors and affiliations

Mladen Vassilev-Missana

*5 V. Hugo Str., Sofia–1124, Bulgaria*

Peter Vassilev

*Institute of Biophysics and Biomedical Engineering*

### Abstract

The paper is a continuation of [1]. The considerations are over the class of multiplicative functions with strictly positive values and more precisely, over the pairs (*f*, *g*) of such functions, which have a special property, called in the paper property **S**. For every such pair(*f*, *g*) and for every composite number *n* > 1, the problem of finding the maximum and minimum of the numbers *f*(*d*)*g*(*n/d*), when *d* runs over all proper divisors of n, is completely solved. Since some classical multiplicative functions like Euler’s totient function *φ*, Dedekind’s function *ψ*, the sum of all divisors of m, i.e. *σ(m)*, the number of all divisors of* m*, i.e. *τ(m)*, and 2^{ω(m)} (where *ω(m)* is the number of all prime divisors of m) form pairs having property **S**, we apply our results to these functions and also resolve the questions of finding the maximum and minimum of the numbers *φ(d)σ(n/d)*, *φ(d)ψ(n/d)*, *τ(d)σ(n/d)*, 2* ^{ω(d)}σ(n/d)*, where

*d*runs over all proper divisors of

*n*. In addition some corollaries from the obtained results, concerning unitary proper divisors, are made. Since many other pairs of multiplicative functions (except the considered in the paper) have property

**S**, they may be investigated in similar manner in a future research.

### Keywords

- Multiplicative functions
- Divisors
- Proper divisors
- Unitary divisors
- Proper unitary divisors
- Prime numbers
- Composite number

### AMS Classification

- 11A25

### References

- Vassilev-Missana, V. Some Results on Multiplicative Functions. Notes on Number Theory and Discrete Mathematics, Vol. 16, 2010, No. 4, 29-40.
- Weisstein, Eric W. “Unitary Divisor.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/UnitaryDivisor.html
- Sondow, Jonathan andWeisstein, Eric W. “Bertrand’s Postulate.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BertrandsPostulate.html
- Nagura, J. “On the interval containing at least one prime number”. Proc. Japan Acad. Vol. 28, 1952, No.4, 177-181

## Related papers

## Cite this paper

APAVassilev-Missana, M., & Vassilev, P. (2011). New results on some multiplicative functions. Notes on Number Theory and Discrete Mathematics, 17(2), 18-30.

ChicagoVassilev-Missana, Mladen, and Peter Vassilev. “New Results on Some Multiplicative Functions.” Notes on Number Theory and Discrete Mathematics 17, no. 2 (2011): 18-30.

MLAVassilev-Missana, Mladen, and Peter Vassilev. “New Results on Some Multiplicative Functions.” Notes on Number Theory and Discrete Mathematics 17.2 (2011): 18-30. Print.