On the denseness of certain reciprocal power sums

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. For any integer $n\ge 1$, one lets $\mathcal{S}_n: = \{s_1, ..., s_n\}$ and $H_f(\mathcal{S}_n): = \sum_{k = 1}^{n}\frac{1}{f(k)^{s_{k}}}$. In this paper, 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 $\alpha_{x^2+1} = \frac{1}{2}\big(\pi \frac{e^{2\pi}+1}{e^{2\pi}-1}-1\big)\approx 1.076674$. Our dense result infers that 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 $1-\varepsilon < H_{x^2+1}(\mathcal{S}^{(1)}_{n_1}) < 1$ and < $H_{x^2+1}(\mathcal{S}^{(2)}_{n_2}) < 1+\varepsilon$. Finally, we conjecture that for any polynomial $f(x)$ of integer coefficients satisfying that $f(m)\ne 0$ for any positive integer $m$ and for any infinite sequence $\mathcal{S} = \{s_i\}_{i = 1}^\infty$ of positive integers (not necessarily increasing and not necessarily distinct), there is a positive integer $N$ such that for any integer $n$ with $n\ge N$, $H_f(\mathcal{S}_n)$ is not an integer. Particularly, we guess that for any positive integer $n$, $H_{x^2+1}(\mathcal{S}_n)$ is never equal to 1.