Extremality Properties of Some Diophantine Series

We study the convergence properties of the series Ψs(α) := ∑ n≥1 ||n2α|| ns+1||nα|| with respect to the values of the real numbers α and s, where ||x|| is the distance of x to Z. For example when s ∈ (0, 1], the convergence of Ψs(α) strongly depends on the diophantine nature of α, mainly its irrationality exponent. We also conjecture that Ψs(α) is minimal at √ 5 for s ∈ (0, 1] and we present evidences in favor of that conjecture. For s = 1, we formulate a more precise conjecture about the value of the abscissa uk where the Fk-partial sum of Ψ1(α) is minimal, Fk being the k-th Fibonacci number. A similar study it made for the partial sums of the series Ψ̃1(α) := ∑ n≥1(−1) ||n 2α|| n2||nα|| that we conjecture to be minimal at √ 2/2.