Niveau: Supérieur, Master, Bac+5
CONVERGENCE AND MODULAR TYPE PROPERTIES OF A TWISTED RIEMANN SERIES T. RIVOAL AND J. ROQUES Abstract. We consider the series ?(?) = ∑∞ m=1 1 m2 sin(2pim 2?) cot(pim?), 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 ?, which is almost everywhere but not everywhere convergent. We first prove that ? enjoys a modular type property, in the following sense (with ?n the n-th partial sum of ?): For all ? ? (0, 1], the sequence ?N (?) ? ??b?Nc(?1/?) has a finite simple limit ?(?) as N ? +∞. Using analytic properties of ?, we then prove that ?(?) converges if and only if ? is irrational and ∑ j log(qj+1)/qj converges (Brjuno's condition), where qj is the j-th denominator in the sequence of convergents to ?. This completes the results obtained in the above mentioned paper, where it was proved that ?(?) converges absolutely under Brjuno's condition.
- brjuno's condition
- fraction expansion
- riemann series
- j?1 ∑
- since ? ?
- see section
- sin
- functions
- convergence properties