A p-adic Study of the Partial Sums of the Harmonic Series

David W. Boyd

Abstract

Let H_{n} = 1 + \frac{1}{2} + ⋯ + \frac{1}{n} be the n-th partial sum of the harmonic series. A classical result of Wolstenholme states that, if p > 3 is prime, the numerator of H_{p−1} is divisible by p^{2}. Here we consider, for a given prime p, the set J_{p} of n for which p divides the numerator of H_{n}. This set J_{p} had been previously determined for p = 2, 3, 5, 7. One of our results is that J_{11} contains exactly 638 integers, the largest of which is a number of 31 decimal digits. We determine J_{p} for all p < 550 with three exceptions: 83, 127 and 397.

The computation is based on a new p-adically convergent formula for the quantity H_{pn} −H_{n}/p. We describe a probabilistic model for the sets J_{p}, based on branching processes. The model predicts that |J_{p}| = O(p^{2}(log log p)^{2+ϵ}), and that there are infinitely many p with |J_{p}| ≥ p^{2}(log log p)^{2}. This strengthens an earlier conjecture of Eswarathasan and Levine that |J_{p}| is finite for all p. Another prediction of the model is that there will be infinitely many pairs (n, p) for which p^{3} divides the numerator of H_{n}, but only finitely many for which p^{4} divides H_{n}.

It has been conjectured that there are infinitely many p for which |J_{p}| = 3. We give a probabilistic argument that suggests that such primes have a density 1/e in the set of all primes, and experimentally confirm this by a determination of all such p ≤ 10^{5}.