# A certain reciprocal power sum is never an integer

Research paper by **Junyong Zhao, Shaofang Hong, Xiao Jiang**

Indexed on: **20 Dec '18**Published on: **20 Dec '18**Published in: **arXiv - Mathematics - Number Theory**

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#### 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 Erd\H{o}s, 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 $\bigcup\limits_{\mathcal{S}\in
(\mathbb{Z}^+)^{\infty}} \{ H_f(\mathcal{S}_n) | n\in \mathbb{Z}^+ \}$ is dense
in the interval $(0,\alpha_f)$ with
$\alpha_f:=\sum_{k=1}^{\infty}\frac{1}{f(k)}$. It is well known that $\alpha_f=
\frac{1}{2}\big(\pi \frac{e^{2\pi}+1}{e^{2\pi}-1}-1\big)\approx 1.076674$ when
$f(x)=x^2+1$. Our dense result infers that when $f(x)=x^2+1$, for any
sufficiently small $\varepsilon >0$, there are positive integers $n_1$ and
$n_2$ and infinite sequences $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(2)}$ of
positive integers such that $1-\varepsilon<H_f(\mathcal{S}^{(1)}_{n_1})<1$ and
$1<H_f(\mathcal{S}^{(2)}_{n_2})<1+\varepsilon$.