AIMS Mathematics, 2019, 4(3): 412-419. doi: 10.3934/math.2019.3.412

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On the denseness of certain reciprocal power sums

Mathematical College, Sichuan University, Chengdu 610064, P. R. China

By $(\mathbb{Z}^+)^{\infty}$ we denote the set of all theinfinite sequences $\mathcal{S}=\{s_i\}_{i=1}^{\infty}$ of positiveintegers (note that all the $s_i$ are not necessarily distinct and notnecessarily monotonic). Let $f(x)$ be a polynomial of nonnegativeinteger 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 thatif $\frac{1}{f(k)}\le\sum_{i=1}^\infty\frac{1}{f(k+i)}$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 densein 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$1<H_{x^2+1}(\mathcal{S}^{(2)}_{n_2})<1+\varepsilon$.Finally, we conjecture that for any polynomial $f(x)$of integer coefficients satisfying that $f(m)\ne 0$ for anypositive 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.
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