The denominators of harmonic numbers (Revised)

Peter Shiu

paper

The denominators of harmonic numbers (Revised)

arXiv:1607.02863

Abstract

The denominators $d_{n}$ of the harmonic number $1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ do not increase monotonically with~ $n$ . It is conjectured that $d_{n} = D_{n} = LCM (1, 2, \dots, n)$ infinitely often. For an odd prime $p$ , the set ${n : p d_{n} ∣ D_{n}}$ has a harmonic density. Moreover, for $2 < p_{1} < p_{2} < \dots < p_{k}$ , with $lo g p_{1} / lo g p_{i}$ ( $1 \leq i \leq k$ ) being linearly independent, there exists $n$ such that $p_{1} p_{2} \dots p_{k} d_{n} ∣ D_{n}$ .

7 pages