A certain reciprocal power sum is never an integer

Abstract

By $({\mathbb{Z}}^{+}{)}^{\infty }$ we denote the set of all the infinite sequences $\mathcal{S}=\{s_i{\}}_{i=1}^{\infty }$ of positive integers (note that all the $s_i$ are not necessarily distinct and not necessarily monotonic). Let $f(x)$ be a polynomial of nonnegative integer coefficients. Let ${\mathcal{S}}_n:=\{s_1,...,s_n\}$ and $H_f({\mathcal{S}}_n):={\sum }_{k=1}^n\frac{1}{f(k{)}^{s_k}}$. When $f(x)$ is linear, Feng, Hong, Jiang and Yin proved in [A generalization of a theorem of Nagell, Acta Math. Hungari, in press] that for any infinite sequence $\mathcal{S}$ of positive integers, $H_f({\mathcal{S}}_n)$ is never an integer if $n\ge 2$. Now let deg$f(x)\ge 2$. Clearly, $0<H_f({\mathcal{S}}_n)<\zeta (2)<2$. But it is not clear whether the reciprocal power sum $H_f({\mathcal{S}}_n)$ can take 1 as its value. In this paper, with the help of a result of Erdős, we use the analytic and $p$-adic method to show that for any infinite sequence $\mathcal{S}$ of positive integers and any positive integer $n\ge 2$, $H_f({\mathcal{S}}_n)$ is never equal to 1. Furthermore, we use a result of Kakeya to show that if $\frac{1}{f(k)}\le {\sum }_{i=1}^{\infty }\frac{1}{f(k+i)}$ holds for all positive integers $k$, then the union set $\bigcup _{\mathcal{S}\in ({\mathbb{Z}}^{+}{)}^{\infty }}\{H_f({\mathcal{S}}_n)|n\in {\mathbb{Z}}^{+}\}$ is dense in the interval $(0,{\alpha }_f)$ with ${\alpha }_f:={\sum }_{k=1}^{\infty }\frac{1}{f(k)}$. It is well known that ${\alpha }_f=\frac{1}{2}(\pi \frac{e^{2\pi }+1}{e^{2\pi }-1}-1)\approx 1.076674$ when $f(x)=x^2+1$. Our dense result infers that when $f(x)=x^2+1$, for any sufficiently small $\epsilon >0$, there are positive integers $n_1$ and $n_2$ and infinite sequences ${\mathcal{S}}^{(1)}$ and ${\mathcal{S}}^{(2)}$ of positive integers such that $1-\epsilon <H_f({\mathcal{S}}_{n_1}^{(1)})<1$ and $1<H_f({\mathcal{S}}_{n_2}^{(2)})<1+\epsilon$.

11 pages