Fractal Curves from Prime Trigonometric Series

We study the convergence of the parameter family of series: V-alpha,V- beta(t) = Sigma(p) p(-alpha) exp(2 pi ip(beta)t), alpha, beta is an element of R->0, t is an element of [0,1) defined over prime numbers p and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of alpha, beta is analyzed in terms of Holder 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.