On the periodicity of an arithmetic function associated to the Euler phi function

Details On the periodicity of an arithmetic function associated to the Euler phi function On the periodicity of an arithmetic function associated to the Euler phi function

2012-01-04

arxiv.org/abs/1201.0931v1

Mathematics

Number Theory

15 pages

Scientific paper

Let $k\ge 1,a\ge 1,b\ge 0$ and $ c\ge 1$ be integers. Let $f$ be a multiplicative function with $f(n)\ne 0$ for all $n\in\mathbb{N}^*$. We define the arithmetic function $g_{k,f}$ for any positive integer $n$ by $g_{k,f}(n):=\frac{\prod_{i=0}^k f(b+a(n+ic))} {f({\rm lcm}_{0\le i\le k} \{b+a(n+ic)\})}$. We first show that $g_{k,f}$ is periodic and $cL_k$ is its period. Consequently, we provide detailed $p$-adic analysis to the periodic function $g_{k,\varphi}$, and finally determine the smallest period of $g_{k,\varphi}$, where $\varphi$ is the Euler phi function.

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