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

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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.
  Article Metrics

Keywords denseness; infinite series; reciprocal power sum; convergence

Citation: Xiao Jiang, Shaofang Hong. On the denseness of certain reciprocal power sums. AIMS Mathematics, 2019, 4(3): 412-419. doi: 10.3934/math.2019.3.412


  • 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.    


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved