File:Comparison of prime-counting approximations.svg
Summary
| Description |
English: Plot comparing three functions related to the distribution of prime number: the prime-counting function π(x), the logarithmic integral Li(x), and the aproximation x/log(x).
Português: Gráfico comparando três funções relacionadas à distribuição de números primos: a função de contagem de números primos π(x), a integral logarítmica Li(x), e a aproximação x/log(x). |
| Date | |
| Source |
Own work Category:Unspec SVG created with Matplotlib#00041122Comparison%20of%20prime-counting%20approximations.svg |
| Author | Vickvvy |
Matplotlib (Python)
# Author: Vickvvy, 2025-05-19
import matplotlib.pyplot as plt
from sympy import primepi
from scipy.integrate import quad
import numpy as np
def xlogx(x):
return x / np.log(x)
def Li_array(x_array):
return np.array([quad(lambda t: 1 / np.log(t), 2, xi)[0] for xi in x_array])
x = np.linspace(2.01, 10000, 500000)
y1 = xlogx(x)
y2 = Li_array(x)
y3 = [primepi(int(xi)) for xi in x]
plt.figure(figsize=(10, 6))
plt.plot(x, y1, label=r"$\frac{x}{\log x}$", color='red')
plt.plot(x, y2, label=r"$\mathrm{Li}(x)$", color='blue')
plt.plot(x, y3, label=r"$\pi(x)$", color='green')
plt.legend()
plt.xlabel("x")
plt.ylabel("y")
plt.grid(True)
plt.savefig("pi_li_xlogx.svg", format='svg')
plt.show()
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