Multiple harmonic sums and multiple harmonic star sums are (nearly) never integers

TL;DR

This paper investigates the integrality of multiple harmonic sums and their star variants, demonstrating that these sums are almost never integers, with only a few known exceptions, thus extending previous results on harmonic sums.

Contribution

It generalizes the non-integer property to a broader class of multiple harmonic sums and star sums, establishing that they are nearly never integers beyond known special cases.

Findings

Multiple harmonic sums are almost never integers.

The paper extends known results to star sums.

Only finitely many exceptions exist, which are explicitly characterized.

Abstract

It is well known that the harmonic sum $H_n(1)=\sum_{k=1}^n\frac{1}{k}$ is never an integer for $n>1$. In 1946, Erd\H{o}s and Niven proved that the nested multiple harmonic sum $H_n(\{1\}^r)=\sum_{1\le k_1<\dots<k_r\le n}\frac{1}{k_1\cdots k_r}$ can take integer values only for a finite number of positive integers $n$. In 2012, Chen and Tang refined this result by showing that $H_n(\{1\}^r)$ is an integer only for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In this paper, we consider the integrality problem for arbitrary multiple harmonic and multiple harmonic star sums and show that none of these sums is an integer with some natural exceptions like those mentioned above.