The maximal order of a class of multiplicative arithmetical functions

Abstract

We prove simple theorems concerning the maximal order of a large class of multiplicative functions. As an application, we determine the maximal orders of certain functions of the type $\sigma_A(n)= \sum_{d\in A(n)} d$, where A(n) is a subset of the set of all positive divisors of $n$, including the divisor-sum function $\sigma(n)$ and its unitary and exponential analogues. We also give the minimal order of a new class of Euler-type functions, including the Euler-function $\phi(n)$ and its unitary analogue.