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## 基础数学研究中心2020年系列报告8期（数论）

Let $n$ be a positive integer. In 1875,H.J.S.Smith published his renowned theorem stating that the determinant of the $n\times n$ matrix $(\gcd(i, j))$ having the greatest common divisor of $i$ and $j$ as its $(i, j)$-entry is equal to the product $\prod_{k=1}^n\varphi (k)$, where $\varphi$ stands for the Eulers totient function. In this talk, we will first present the proof of this beautiful result, and then discuss some related results and problems.