The shortest harmonic sums with decreasing denominator

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 $$\underset{a\to \infty }{\mathrm{liminf}}\left(\frac{b(a)-a}{\log a}\right)$$ exists and is positive. Here we find its exact value.

9 pages