On the integrality of the first and second elementary symmetricfunctions of $1, 1/2^{s_2}, ...,1/n^{s_n}$

Abstract

It is well known that the harmonic sum $H_{n}(1)=\sum_{1\leq k\leq n}\frac{1}{k}$is never an integer for $n>1$. Erd\"{o}s and Niven proved in 1946 thatthe multiple harmonic sum$H_{n}(\{1\}^r)=\sum_{1\leq k_{1}<\cdots< k_{r}\leq n}\frac{1}{k_{1}\cdots k_{r}}$can take integer values for at most finite many integers $n$. In 2012, Chenand Tang refined this result by showing that $H_{n}(\{1\}^r)$ is an integeronly for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In this paper, we consider theintegrality problem for the first and second elementary symmetric functionof $1, 1/2^{s_2}, ...,$ $1/n^{s_n}$, we show that none of themis an integer with some natural exceptions.

Keywords

• elementary symmetric function| integrality| Bertrand’s postulate| <em>p</em>-adic valuation