On the p-adic valuation of harmonic numbers

For any prime number p  , let JpJp be the set of positive integers n such that p divides the numerator of the n  -th harmonic number HnHn. An old conjecture of Eswarathasan and Levine states that JpJp is finite. We prove that for x≥1x≥1 the number of integers in Jp∩[1,x]Jp∩[1,x] is less than 129p2/3x0.765129p2/3x0.765. In particular, JpJp has asymptotic density zero. Furthermore, we show that there exists a subset SpSp of the positive integers, with logarithmic density greater than 0.273, and such that for any n∈Spn∈Sp the p  -adic valuation of HnHn is equal to −⌊logp⁡n⌋−⌊logp⁡n⌋.