On the non-monotonicity of the denominator of generalized harmonic sums

TL;DR

This paper investigates the non-monotonic behavior of denominators in generalized harmonic sums, proving that for fixed starting point, the denominator can decrease at certain points and establishing bounds on where this occurs.

Contribution

The paper extends the study of harmonic sum denominators to a more general setting, affirmatively answers Erdős and Graham's question, and provides explicit bounds on the least such index.

Findings

Denominator of generalized harmonic sums can decrease at specific points.

Established bounds for the index where denominator decreases occur.

Confirmed the existence of such points for all large enough starting values.

Abstract

Let $\displaystyle \sum_{i=a}^b \frac{1}{i} = \frac{u_{a,b}}{v_{a,b}}$ with $u_{a,b}$ and $v_{a,b}$ coprime. Erd\H{o}s and Graham asked the following: Does there, for every fixed $a$, exist a $b$ such that $v_{a,b} < v_{a,b-1}$? If so, what is the least such $b = b(a)$? In this paper we will investigate these problems in a more general setting, answer the first question in the affirmative and obtain the bounds $a + 0.54\log(a) < b(a) \le 4.374(a-1)$, which hold for all large enough $a$.