Über trigonometrische Reihen mit zahlentheoretischen Koeffizienten. (On some trigonometric series with number theoretic coefficients)
The authors study the series \(T(\alpha)=\sum_{p prime}\frac{\sin (2\pi p\alpha)}{p}\). It is shown that it is uniformly convergent for \(\alpha\in {\mathbb{R}}\). The function T is nondifferentiable on a noncountable set of measure zero and nowhere of bounded variation. It is highly probable that T is nowhere differentiable, but this seems to be a hard problem. In the proofs estimates for exponential sums with primes (Vinogradov's bound) and the theorem of Siegel-Walfisz are used. The main ideas are similar to those introduced by the authors in two earlier papers [Math. Z. 149, 155- 167 (1976; Zbl 0312.10026); Acta Math. Acad. Sci. Hung. 33, 263-288 (1979; Zbl 0419.10036)].
scientific article; zbMATH DE number 4200304