# Fractal curves from prime trigonometric series

## TL;DR

This paper investigates the convergence, fractal properties, and randomness of a family of prime-based trigonometric series, revealing their fractal nature and connections to random walks through analytical and numerical methods.

## Contribution

It introduces a detailed analysis of the convergence and fractal characteristics of prime trigonometric series, linking them to random walk behaviors and providing numerical insights.

## Key findings

- Series exhibit fractal graphs with self-similarity.
- Convergence depends on parameters nd eta.
- Series show links to random walk properties.

## Abstract

We study the convergence of the parameter family of series $$V_{\alpha,\beta}(t)=\sum_{p}p^{-\alpha}\exp(2\pi i p^{\beta}t),\quad \alpha,\beta \in \mathbb{R}_{>0},\; t \in [0,1)$$ 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 H\"older continuity, self similarity and fractal dimension, backed with numerical results. We also discuss the link of this series to random walks and consequently, explore numerically its random properties.