Extremality Properties of Some Diophantine Series

Abstract

We study the convergence properties of the series $Ψ_s(α) := \sum_{n≥1} \frac{||n^2α||} {n^{s+1}||nα||}$ with respect to the values of the real numbers $α$ and $s$, where $||x||$ is the distance of $x$ to $\mathbb{Z}$. For example, when $s ∈ (0, 1]$, the convergence of $Ψ_s(α)$ strongly depends on the Diophantine nature of $α$, mainly its irrationality exponent. We also conjecture that $Ψ_s(α)$ is minimal at $\sqrt{5}$ for $s ∈ (0, 1]$, and we present evidence in favor of that conjecture. For $s = 1$, we formulate a more precise conjecture about the value of the abscissa $u_k$ where the $F_k$-partial sum of $Ψ_1(α)$ is minimal, $F_k$ being the $k$th Fibonacci number. A similar study is made for the partial sums of the series $\tilde{Ψ}_1(α) := \sum_{n≥1}(−1)^n \frac{||n^2α||} {n^2||nα||}$, which we conjecture to be minimal at $\sqrt{2}/2$.