Maximal order of a class of multiplicative functions

Abstract. In this paper we obtain the maximal order of the multiplicative function given at the prime powers by \(f(p^k) = \exp\{ h(k)l(p)\}\) where \(h(\cdot)\) and \(l(\cdot)\) are increasing and decreasing functions respectively with \(l(p)\) regularly varying of index \(-\alpha\ \) \((0\le \alpha < 1)\). For example, we show that under appropriate conditions \[ \max_{n\le N} \log f(n) \sim \biggl(\sum_{n=1}^{\infty} \Delta h(n)^{1/\alpha}\biggr)^{\alpha}L(\log N) \] where \(L(x) = \sum_{p\le x} l(p)\) and \(\Delta h(n) = h(n)-h(n-1)\).