On the p-adic valuation of harmonic numbers

For any prime number $p$, let $J_p$ be the set of positive integers $n$ such that $p$ divides the numerator of the $n$-th harmonic number $H_n$. An old conjecture of Eswarathasan and Levine states that $J_p$ is finite. We prove that for $x \geq 1$ the number of integers in $J_p \cap [1,x]$ is less than $129p^{2/3} x^{0.765}$. In particular, $J_p$ has asymptotic density zero. Furthermore, we show that there exists a subset $S_p$ of the positive integers, with logarithmic density greater than $0.273$, and such that for any $n \in S_p$ the $p$-adic valuation of $H_n$ is equal to $- \lfloor \log_p n \rfloor$.