# Fractal Curves from Prime Trigonometric Series

^{1}

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## Abstract

**:**

## 1. Introduction

**Remark**

**1.**

## 2. Convergence and Differentiability

- The faster the coefficients ${a}_{k}$ decrease for $k\to \infty $, the smaller is the influence of the higher frequencies. This implies that the series converges better and the resulting function is smoother.
- The faster the frequencies ${n}_{k}$ increase or, equivalently, the greater the gaps, the smaller the period of the oscillation becomes, so that one obtains more peaks and sinks in one interval, which increases the fractal character.

#### 2.1. Historical Remarks

**Theorem**

**1**

**Theorem**

**2**

**.**1. If $0<\alpha \le \frac{1}{2}$, then the series is not a Fourier series of an ${L}^{1}$-function. If $0<\alpha <\frac{1}{2}$, then ${R}_{\alpha}$ converges at x if and only if $x=\frac{a}{q}$, where $a,q$ are coprime and four divides $q-2$.

#### 2.2. Preliminary Definitions

**Proposition**

**1**(Jaffard)

**.**

#### 2.3. Differentiability of ${V}_{\alpha ,\beta}$

**Proposition**

**2.**

**Proof.**

**Theorem**

**3.**

- Then, the series ${V}_{\alpha ,\beta}(n,t)$ converges uniformly and absolutely to a continuous function ${V}_{\alpha ,\beta}(t)$.
- For $m\ge 1$, if further $\alpha -m\beta >1$, then the function ${V}_{\alpha ,\beta}(t)$ is ${C}^{m}$, i.e., m-times continuously differentiable.

**Proof.**

**Remark**

**2.**

**Theorem**

**4.**

**Proof.**

**Remark**

**3.**

#### 2.4. Self-Similarity and Fractal Dimension

#### Fractal Dimension of ${V}_{\alpha ,\beta}$

**Remark**

**4.**

## 3. Random Properties for ${\mathit{V}}_{\mathit{\alpha},\mathit{\beta}}$

#### 3.1. Lacunary Sequences Behaving as Independent Random Variables: Short Overview

**Theorem**

**5**

**.**Assume the Hadamard gap condition. Assume further that ${A}_{N}:=\sqrt{\frac{1}{2}{\sum}_{k=1}^{N}{a}_{k}^{2}}\to \infty $ and there exists $\delta >0$ such that ${\mathrm{lim}}_{N\to \infty}\frac{{a}_{N}}{{A}_{N}^{1-\delta}}=0$. Then, without changing the distribution of the process:

#### 3.2. The Central Limit Theorem

## 4. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Graph of (

**a**): $\mathrm{sin}(2\pi {n}_{k}x)$ with ${n}_{k}={2}^{4}$ and ${n}_{k}={2}^{5};$ and (

**b**): $\mathrm{sin}(2\pi nx)$ with $n=5$ and $n=6$ (displayed for $x\in [0,1)$ at ${10}^{5}$ points).

**Figure 2.**Graph of (

**a**):${\sum}_{n=1}^{10}\mathrm{sin}({2}^{n}\pi x)$; (

**b**): ${\sum}_{n=1}^{1000}\mathrm{sin}(2\pi nx)$; (

**c**): ${\sum}_{p\le 1000}\mathrm{sin}(2\pi px)$ (displayed for $x\in [0,1)$ at ${10}^{4}$ points).

**Figure 3.**Graph of ${V}_{1.5,2}({10}^{5},t)$ at $5\times {10}^{4}$ discrete points in each direction (interpolated).

**Figure 4.**Graph of ${V}_{1.5,1.5}({10}^{5},t)$ at $5\times {10}^{4}$ discrete points in each direction (interpolated).

**Figure 5.**Graph of ${V}_{1.5,1}({10}^{5},t)$ at $5\times {10}^{4}$ discrete points in each direction (interpolated).

**Figure 7.**The graph of the real part of $-\frac{1}{2}{V}_{1,1}({10}^{6},t)$ (in black) and ${V}_{1,1}({10}^{6},t/3)+\frac{1}{3}$ (in green).

**Figure 8.**(

**a**): Box dimension for the graph of ${V}_{\alpha ,\beta}$ (computed at ${10}^{5}$ points) in dependence of the fraction of the powers $\frac{\alpha}{\beta}$ with $\alpha \in [1,1.5]$ and $\beta \in [0.5,3]$. Remark that ${V}_{\alpha ,\beta}$ is not convergent for $\alpha =1$. (

**b**): example for $\alpha =1.5,\beta =2$ to show how the box dimension was numerically approximated.

**Figure 9.**Normal distribution of $\frac{1}{N}{\sum}_{k=1}^{N}\mathrm{sin}({p}_{k}x)$ for x uniformly distributed in $\left[-\pi ,\pi \right]$.

**Figure 10.**Normal distribution of $\frac{1}{N}{\sum}_{k=1}^{N}\mathrm{sin}(\pi {p}_{k}^{\frac{3}{2}}x)$ for x uniformly distributed in $\left[-\pi ,\pi \right]$.

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**MDPI and ACS Style**

Vartziotis, D.; Bohnet, D. Fractal Curves from Prime Trigonometric Series. *Fractal Fract.* **2018**, *2*, 2.
https://doi.org/10.3390/fractalfract2010002

**AMA Style**

Vartziotis D, Bohnet D. Fractal Curves from Prime Trigonometric Series. *Fractal and Fractional*. 2018; 2(1):2.
https://doi.org/10.3390/fractalfract2010002

**Chicago/Turabian Style**

Vartziotis, Dimitris, and Doris Bohnet. 2018. "Fractal Curves from Prime Trigonometric Series" *Fractal and Fractional* 2, no. 1: 2.
https://doi.org/10.3390/fractalfract2010002