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

Authors:	J M De Koninck I Kátai

Institution:	(1) Université Laval, Québec, Canada;(2) Eötvös Loránd University, Budapest, Hungary

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 (1)=(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 running over shifted primes.Research supported in part by a grant from NSERC.Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA.

Keywords:	2000 Mathematics Subject Classifications: 11A25 11N25 11N37
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