On the denominators of signed harmonic sums

Source abstract

Let $\eps=(\eps_i)_{i\ge1}$ be a sequence with $\eps_i\in\{-1,1\}$, and let $v_n(\eps)$ be the positive denominator of $\sum_{i=1}^n\eps_i/i$ in lowest terms. We prove that $|\{1\le n\le x:v_n(\eps)\ne v_{n+1}(\eps)\}|=o(x)$ uniformly in $\eps$. In particular, the set of positive integers $n$ with $v_n(\eps)=v_{n+1}(\eps)$ has asymptotic density one for every sequence of signs. This answers a problem of Wu and Chen, \emph{J. Number Theory} \textbf{200} (2019), 397--406.