Fractal Curves from Prime Trigonometric Series

[摘要] We study the convergence of the parameter family of series:Vα , β( t )=∑ p p− α exp( 2 π ip βt ) , 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.

[学科分类] 数值分析

[关键词] trigonometric series;lacunary series;Hölder continuity;fractality;r;om Fourier series;prime distribution