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- 2018
Fractal Curves from Prime Trigonometric SeriesDOI: https://doi.org/10.3390/fractalfract2010002 Keywords: trigonometric series, lacunary series, H?lder continuity, fractality, random Fourier series, prime distribution Abstract: Abstract 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 H?lder 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. View Full-Tex
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