Convergence and Modular Type Properties of a Twisted Riemann Series

We consider the series Φ(α) = ∞ m=1 m 1 2 sin(2πm2α) 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 ap- peared 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) − αΦαN(−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 j log(q j+1)/q j converges (Brjuno’s condition), where q j is the j-th denominator in the sequence of convergents to α. This completes the results obtained in the above mentioned paper, where it was proved that Φ(α) converges absolutely under Brjuno’s condition.