On two conjectures concerning the partial sums of the harmonic series

Abstract:

Let ${S_n}$ denote the $n$th partial sum of the harmonic series. For a given positive integer $k > 1$, there exists a unique integer ${n_k}$ such that ${S_{{n_k} - 1}} < k < {S_{{n_k}}}$. It has been conjectured that ${n_k}$ is equal to the integer nearest ${e^{k - y}}$, where $\gamma$ is Euler’s constant. We provide an estimate on ${n_k}$ which suggests that this conjecture may have to be modified. We also propose a conjecture concerning the amount by which ${S_{{n_k} - 1}}$ and ${S_{{n_k}}}$ differ from $k$.