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

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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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# References

1. Y. G. Chen and M. Tang, On the elementary symmetric functions of 1, 1/2, ..., 1/n, Am. Math. Mon, 119 (2012), 862-867.

2. P. Erdös and I. Niven, Some properties of partial sums of the harmonic series, B. Am. Math. Soc., 52 (1946), 248-251.

3. Y. L. Feng, S. F. Hong, X. Jiang, et al. A generalization of a theorem of Nagell, Acta Math. Hung., 157 (2019), 522-536.

4. S. F. Hong and C. L. Wang, The elementary symmetric functions of reciprocals of the elements of arithmetic progressions, Acta Math. Hung., 144 (2014), 196-211.

5. S. Kakeya, On the set of partial sums of an infinite series, Proceedings of the Tokyo Mathematico-Physical Society. 2nd Series, 7 (1914), 250-251.

6. K. Kato, N. Kurokawa, T. Saito, et al. Number theory: Fermat's dream, Translated from the 1996 Japanese original by Masato Kuwata. Translations of Mathematical Monographs, Vol. 186. Iwanami Series in Modern Mathematics, American Mathematical Society, 2000.

7. Y. Y. Luo, S. F. Hong, G. Y. Qian, et al. The elementary symmetric functions of a reciprocal polynomial sequence, C. R. Math., 352 (2014), 269-272.

8. T. Nagell, Eine Eigenschaft gewissen Summen, Skr. Norske Vid. Akad. Kristiania, 13 (1923), 10-15.

9. L. Theisinger, Bemerkung über die harmonische Reihe, Monatsh. Math., 26 (1915), 132-134.

10. C. L. Wang and S. F. Hong, On the integrality of the elementary symmetric functions of 1; 1/3, ..., 1/(2n-1), Math. Slovaca, 65 (2015), 957-962.

11. W. X. Yang, M. Li, Y. L. Feng, et al. On the integrality of the first and second elementary symmetric functions of $1,1/2^{s_2} ,..., 1/n^{s_n}$, AIMS Mathematics, 2 (2017), 682--691

12. Q. Y. Yin, S. F. Hong, L. P. Yang, et al. Multiple reciprocal sums and multiple reciprocal star sums of polynomials are almost never integers, J. Number Theory, 195 (2019), 269-292.