Convergence and Modular Type Properties of a Twisted Riemann Series

Abstract. We consider the series Φ(α) = ∑∞ m=1 1 m2 sin(2πm α) cot(πmα), a twist of the famous continuous but almost nowhere differentiable sine series defined by Riemann. In a slightly different but equivalent form, this series appeared in the first author’s paper [On the distribution of multiple of real numbers, Monatsh. Math 164.3 (2011), 325–360]. We pursue here the study of Φ, which is almost everywhere but not everywhere convergent. We first prove that Φ enjoys a modular type property, in the following sense (with Φn the n-th partial sum of Φ): For all α ∈ (0, 1], the sequence ΦN (α) − αΦ⌊αN⌋(−1/α) has a finite simple limit Ω(α) as N → +∞. Using analytic properties of Ω, we then prove that Φ(α) converges if and only if α is irrational and ∑