We study the convergence of the parameter family of series:
defined over prime numbers p
and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of
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.
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