  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 $1/b, 1/(a+b),..., 1/(an-a+b)$ 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 $1/b, 1/(a+b),..., 1/(an-a+b)$ 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.
