Extremality Properties of Some Diophantine Series

Abstract

We study the convergence properties of the series Ψ _s (α) := with respect to the values of the real numbers α and s, where ||x|| is the distance of x to . 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 √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 Ψ₁(α) is minimal, F_k being the kth Fibonacci number. A similar study is made for the partial sums of the series , which we conjecture to be minimal at √2/2.

Keywords:

• Diophantine approximation
• continued fractions
• Dirichlet series