[Solution] Fractional parts of Harmonic numbers

Problem statement here.

The problem was to show that fractional parts of harmonic numbers can get arbitrarily close to zero.


It is known that harmonic numbers are divergent. A quick proof of this can be done using the fact that $\frac{1}{x}$ is the derivate of $\log x$ and applying Rolle's theorem to get:

$$ \frac{1}{n+1} \lt \log(n+1) - \log n \lt \frac{1}{n}$$

Add up and get that harmonic number $H_n$ is at least as big as $\log (n+1) $ and so diverges.

Now let $n_k$ be the smallest integer such that $H_{n_k} \gt k$.

We must have that $H_{n_k - 1} \lt k \lt H_{n_k}$ i.e

$$ H_{n_k} - \frac{1}{n_k} \lt k \lt H_{n_k}$$

Thus the fractional part of $H_{n_k}$ is no more than $\frac{1}{n_k}$. Since $n_k \to \infty$ as $k \to \infty$, the fractional parts can be made arbitrarily small.

$n_k$ grows approximately as $e^{k - \gamma}$ where $\gamma$ is the euler mascheroni constant. It is known that $n_k$ is either $\lfloor e^{k - \gamma} \rfloor$ or  $\lfloor e^{k - \gamma} \rfloor + 1$.

It is also conjectured to be the integer nearest to $e^{k - \gamma} + \frac{1}{2}$, attributed to Comtet (who proved the statement in the previous paragraph).