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 $a$.
Categories: math.NT