Fractal Curves from Prime Trigonometric Series

We study the convergence of the parameter family of series: V α , β ( t ) = ∑ p p − α exp ( 2 π i p β t ) , α , β ∈ R > 0 , t ∈ [ 0 , 1 ) defined over prime numbers p and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of α , β is analyzed in terms of Hölder continuity, self-similarity and fractal dimension, backed with numerical results. Although this series is not a lacunary series, it has properties in common, such that we also discuss the link of this series with random walks and, consequently, explore its random properties numerically