New results on some multiplicative functions

Mladen Vassilev-Missana and Peter Vassilev
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 2, Page 18—30
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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 2ω(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), 2ω(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

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

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Cite this paper

APA

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

Chicago

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

MLA

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

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