Fractional parts of harmonic numbers

The $n^{th}$ harmonic number $H_n$ is defined as

$$H_n = \sum_{k=1}^{n} \dfrac{1}{k} = 1 + \dfrac{1}{2} + \dots + \dfrac{1}{n}$$

It is known that $H_n$ is never an integer except for $n=1$.

The problem in this post is to show that we can get arbitrarily close.

i.e.

Show that there is an ascending sequence of integers $n_1 \lt n_2 \lt n_3 \lt \dots$ such that

$$\lim_{k \to \infty} \{H_{n_k}\} = 0$$

where $\{x\}$ is the fractional part of x. Eg, $\{H_2\} = \frac{1}{2}, \{H_3\} = \frac{5}{6}$

In general it seems like a difficult problem to estimate the fractional parts of $H_n$. So if you got here by googling for information on that, sorry, this blog post won't be of much help.


[Solution]

Achieving 1/lcm

Let $d_n$ be the least common multiple of $1,2,\dots, n$. For eg, $d_4 = 12, d_5 = 60, d_9 = 2520$.

Given an $n \ge 1$, Show that there exist integers $A_1, A_2, \dots A_n$ (not necessarily positive) such that

$$\sum_{k=1}^{n} \dfrac{A_k}{k} = \dfrac{1}{d_n}$$



[Solution]