Shifted values of the largest prime factor function and its average value in short intervals
Volume 143 / 2016
Colloquium Mathematicum 143 (2016), 39-62
MSC: 11K06, 11N37.
DOI: 10.4064/cm6474-12-2015
Published online: 3 December 2015
Abstract
We obtain estimates for the average value of the largest prime factor $P(n)$ in short intervals $[x,x+y]$ and of $h(P(n)+1)$, where $h$ is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_q(n)$ stand for the sum of the digits of $n$ in base $q\ge 2$, we show that if $\alpha $ is an irrational number, then the sequence $(\alpha s_q(P(n)))_{n\in \mathbb {N}}$ is uniformly distributed modulo 1.
