Authors and affiliations
The paper is a continuation of . 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.
- Multiplicative functions
- Proper divisors
- Unitary divisors
- Proper unitary divisors
- Prime numbers
- Composite number
- 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
Cite this paperAPA
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.