Visualizing the Prime Number Theorem with Python：A Comparison of \( \pi(x) \) and \( \frac{x}{\log x} \)

Here’s an example from Analytic Number Theory with a complete solution and visualization in $Python$.

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📚 Problem: Prime Number Theorem Approximation

The Prime Number Theorem (PNT) states that:

$$
\pi(x) \sim \frac{x}{\log x}
$$

Where:

• $( \pi(x) )$ is the number of primes less than or equal to $( x )$.
• $( \frac{x}{\log x} )$ approximates $( \pi(x) )$ for large values of $( x )$.

🎯 Goal:

1. Calculate $( \pi(x) )$ for values up to $( x = 100,000 )$.
2. Compare it with the approximation $( \frac{x}{\log x} )$.
3. Visualize the results using a graph.

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📝 Step 1: Define and Calculate Functions

• pi(x): Count primes less than or equal to $( x )$.
• Approximation: Use $( \frac{x}{\log x} )$.

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🧩 Step 2: Python Code Implementation

# Import required libraries
import numpy as np
import matplotlib.pyplot as plt
from sympy import primepi
from math import log

# Define the range and step size
x_values = np.linspace(10, 100000, 200)  # Range from 10 to 100,000 with 200 points

# Calculate prime counting function pi(x)
pi_values = [primepi(int(x)) for x in x_values]

# Calculate the Prime Number Theorem (PNT) approximation
pnt_approx = [x / log(x) for x in x_values]

# Plot the results
plt.figure(figsize=(10, 6))
plt.plot(x_values, pi_values, label=r'$\pi(x)$ (Prime Count)', color='blue')
plt.plot(x_values, pnt_approx, label=r'$\frac{x}{\log x}$ (PNT Approximation)', linestyle='--', color='red')

# Add title and labels
plt.title("Prime Number Theorem: $\pi(x)$ vs. $\dfrac{x}{\log x}$", fontsize=14)
plt.xlabel("x", fontsize=12)
plt.ylabel("Value", fontsize=12)
plt.legend()
plt.grid(True)

# Show the plot
plt.show()

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📊 Step 3: Explanation and Visualization

✅ Explanation:

1. pi(x): Using primepi() from sympy to compute the exact number of primes up to $( x )$.
2. PNT Approximation: Using the formula $( \frac{x}{\log x} )$.
3. Graph:
   • Blue line: Exact prime counting function.
   • Red dashed line: PNT approximation.

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📈 Step 4: Interpretation of Results

• The graph shows that the approximation $( \frac{x}{\log x} )$ becomes increasingly accurate as $( x )$ increases.
• For smaller values of $( x )$, there is a noticeable gap, but for large $( x )$, the approximation closely tracks $( \pi(x) )$.

📢 Insights:

• The Prime Number Theorem provides a remarkably simple yet powerful approximation for the distribution of prime numbers.
• This visualization helps highlight how the error margin decreases with larger $( x )$.