Journal of the
Korean Mathematical Society JKMS

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).