# The Fractal Nature of an Approximate Prime Counting Function

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## Abstract

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## 1. Introduction

- Create a “sound wave” (or more precisely, the von Mangoldt function) which is noisy at prime number times, and quiet at other times. [...]
- “Listen” (or take Fourier transforms) to this wave and record the notes that you hear (the zeroes of the Riemann zeta function, or the “music of the primes”). Each such note corresponds to a hidden pattern in the distribution of the primes.

## 2. Approximations of the Prime Counting Function

## 3. Polygon Transformations and Fourier Polygons

## 4. Deriving Prime Fractals

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**Prime fractal curve ${F}_{c}$ for $n={10}^{6}$ and ${10}^{7}$ evaluation points ${t}_{k}\in (-\pi ,\pi ]$. Boxes of successive zooms are marked red. (

**a**) Full fractal; (

**b**) Zoom level 1; (

**c**) Zoom level 2.

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**MDPI and ACS Style**

Vartziotis, D.; Wipper, J.
The Fractal Nature of an Approximate Prime Counting Function. *Fractal Fract.* **2017**, *1*, 10.
https://doi.org/10.3390/fractalfract1010010

**AMA Style**

Vartziotis D, Wipper J.
The Fractal Nature of an Approximate Prime Counting Function. *Fractal and Fractional*. 2017; 1(1):10.
https://doi.org/10.3390/fractalfract1010010

**Chicago/Turabian Style**

Vartziotis, Dimitris, and Joachim Wipper.
2017. "The Fractal Nature of an Approximate Prime Counting Function" *Fractal and Fractional* 1, no. 1: 10.
https://doi.org/10.3390/fractalfract1010010