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arXiv · 2609.00104

The shortest harmonic sums with decreasing denominator

Wouter van Doorn
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Abstract

For a positive integer $a$, let $b(a)$ be the smallest integer $b > a$ such that the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b}$ is smaller than the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b-1}$. Recently it was shown that the limit inferior $$\liminf_{a \to \infty} \left(\frac{b(a) - a}{\log a}\right)$$ exists and is positive. Here we find its exact value.

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Wouter van Doorn. 2026-08-31. The shortest harmonic sums with decreasing denominator. https://arxiv.org/abs/2609.00104

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