J. Korean Math. Soc. 2022; 59(2): 299-309
Online first article March 1, 2022 Printed March 1, 2022
https://doi.org/10.4134/JKMS.j210123
Copyright © The Korean Mathematical Society.
Yangzhou University; Nanjing Tech University
Let $k\geqslant 2$ be an integer, $S^k=\{1^k,2^k,3^k,\ldots\}$ and $B=\{b_1,b_2,b_3,\ldots\}$ be an additive complement of $S^k$, which means all sufficiently large integers can be written as the sum of an element of $S^k$ and an element of $B$. In this paper we prove that $$\limsup_{n\rightarrow \infty}\frac{\Gamma\left(2-\frac{1}{k}\right)^{\frac{k}{k-1}}\Gamma\left(1+\frac{1}{k}\right) ^{\frac{k}{k-1}}n^{\frac{k}{k-1}}-b_n}{n} \geqslant \frac{k}{2(k-1)}\frac{\Gamma\left(2-\frac{1}{k}\right)^2}{\Gamma\left(2-\frac{2}{k}\right)},$$ where $\Gamma(\cdot)$ is Euler's Gamma function.
Keywords: Additive complement, Gamma function
MSC numbers: Primary 11B13, 11B75
Supported by: The first author was supported by the Natural Science Foundation of Jiangsu Province of China (No. BK20210784). He was also supported by the foundations of the projects ``Jiangsu Provincial Double--Innovation Doctor Program'' (No. JSSCBS20211023) and ``Golden Phenix of the Green City--Yang Zhou'' to excellent PhD (No. YZLYJF2020PHD051). The second author was supported by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (Grant No. 21KJB110001).
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