Prime Number Distribution

Primes are scattered along the number line in a way that appears random at small scales, but follows strict mathematical laws at large scales.

1. The Prime Counting Function $\pi(x)$

The function $\pi(x)$ counts the number of primes less than or equal to $x$. For example, $\pi(10) = 4$ (the primes are 2, 3, 5, 7).

def count_primes(x):
  # Sieve of Eratosthenes to count
  primes = [True] * (x + 1)
  for p in range(2, int(x**0.5) + 1):
      if primes[p]:
          for i in range(p * p, x + 1, p):
              primes[i] = False
  return sum(primes[2:])

print(f"pi(100) = {count_primes(100)}")
print(f"pi(1000) = {count_primes(1000)}")

2. The Prime Number Theorem (PNT)

The PNT describes the asymptotic distribution of primes. It states that: $\pi(x) \sim \frac{x}{\ln x}$ This means that the density of primes near $x$ is approximately $1/\ln x$.

import numpy as np
import matplotlib.pyplot as plt

def pi_approx(x): return x / np.log(x)

x_vals = np.linspace(10, 1000, 100)
y_pnt = pi_approx(x_vals)

plt.plot(x_vals, y_pnt, label='x / ln x', linestyle='--')
plt.title("The Density of Primes (Asymptotic)")
plt.xlabel("x")
plt.ylabel("pi(x)")
plt.legend()
plt.show()

3. The Riemann Zeta Function

The distribution of primes is perfectly encoded in the Riemann Zeta Function: $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p \in \text{Primes}} \frac{1}{1 - p^{-s}}$

This link, discovered by Euler and expanded by Riemann, connects prime numbers (arithmetic) to complex analysis (calculus over complex numbers).

4. The Twin Prime Conjecture

Twin primes are pairs of primes that differ by 2, such as $(3, 5)$ and $(11, 13)$. The conjecture states that there are infinitely many such pairs.

def get_twin_primes(limit):
  primes = [True] * (limit + 1)
  for p in range(2, int(limit**0.5) + 1):
      if primes[p]:
          for i in range(p * p, limit + 1, p):
              primes[i] = False
  
  prime_list = [p for p in range(2, limit + 1) if primes[p]]
  twins = []
  for i in range(len(prime_list) - 1):
      if prime_list[i+1] - prime_list[i] == 2:
          twins.append((prime_list[i], prime_list[i+1]))
  return twins

print(f"Twin primes up to 200: {get_twin_primes(200)}")

5. Summary Check