On the p-adic valuation of harmonic numbers

For any prime number p, let Jp be the set of positive integers n such that p divides the numerator of the n-th harmonic number Hn. An old conjecture of Eswarathasan and Levine states that Jp is finite. We prove that for x ≥ 1 the number of integers in Jp ∩ [1, x] is less than 129p2/3x0.765. In particular, Jp has asymptotic density zero. Furthermore, we show that there exists a subset Sp of the positive integers, with logarithmic density greater than 0.273, and such that for any n ∈ Sp the p-adic valuation of Hn is equal to −logp n.