Some results on multiplicative functions

Mladen Vassilev-Missana
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 16, 2010, Number 4, Pages 29–40
Full paper (PDF, 237 Kb)

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Mladen Vassilev-Missana
5, V. Hugo Str., Sofia-1124, Bulgaria

Abstract

In the present paper some new results, concerning multiplicative functions with strictly positive values, are obtained. In particular, it is shown that if an ordered pair of such functions (f, g) has a certain property (called in the paper S), then for every fixed positive integer n; the minimal and the maximal elements of the set {f(d)g(n/d) : d runs over all divisors of n} are obtained at least for some unitary divisors of n. For these divisors if the maximum of f(d)g(n/d) is reached for d*; then the minimum is reached for n/d* and vice versa (the main results here are Theorems 1-4). The same investigation is made, but when d runs over the set of all divisors of n different than 1 and n (the main result here is Theorem 5). Also corollaries of the mentioned results are obtained and some particular cases are considered.

Keywords

• Multiplicative functions
• Divisors
• Unitary divisors
• Non-unitary divisors

References

1. Weisstein, Eric W. “Unitary Divisor.” From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/UnitaryDivisor.html
2. Sivaramakrishnan, R. Classical Theory of Arithmetic Functions. New York, Dekker, 1989.