The elementary symmetric functions of reciprocals of the elements of arithmetic progressions

Let $a$ and $b$ be positive integers. In 1946, Erd\H{o}s and Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1\mathrm{/}b,1\mathrm{/}(a+b),\mathrm{.}\mathrm{.}\mathrm{.},1\mathrm{/}(an-a+b)$ are integers. In this paper, we show that for any integer $k$ with $1\le k\le n$, the $k$-th elementary symmetric function of $1\mathrm{/}b,1\mathrm{/}(a+b),\mathrm{.}\mathrm{.}\mathrm{.},1\mathrm{/}(an-a+b)$ is not an integer except that either $b=n=k=1$ and $a\ge 1$, or $a=b=1,n=3$ and $k=2$. This refines the Erd\H{o}s-Niven theorem and answers an open problem raised by Chen and Tang in 2012.