On the Mean Value of the Index of Composition of an Integer

Abstract.

For each integer n ≥ 2, let $\lambda(n)={{\rm log}\, n\over{\rm log}\, \gamma(n)}$ be the index of composition of n, where $\gamma(n)=\prod_{p\vert n}p$. For convenience, we write λ(1) = γ(1) = 1. We obtain sharp estimates for $\sum_{x\le n\le x+\sqrt{x}}\lambda(n)$ and $\sum_{n\le x}\lambda(n)$, as well as for $\sum_{x\le n\le x+\sqrt{x}}1/\lambda(n)$ and $\sum_{n\le x}1/\lambda(n)$. Finally we study the sum of λ running over shifted primes.