How does the video derive the Prime Number Theorem formula $x/\ln (x)$ from prime density?

Conditions

• Uses the asymptotic density approximation $1/\ln (x)$.
• The derivation assumes density is constant over the range for simplification.

Reasoning, step by step

1. State that the number of primes is the area under the density curve.
2. Simplify the area calculation by assuming constant density.
3. Multiply the size x by the density $1/\ln (x)$.
4. Obtain the formula #primes(x) = $x/\ln (x)$.

Example

The video shows a shaded area under $y = 1/\ln (x)$, then writes # primes (x) = size * density, and substitutes to get # primes (x) = $x/\ln (x)$.

Common misconceptions

• Treating the constant-density simplification as a rigorous proof rather than a heuristic derivation.
• Assuming $x/\ln (x)$ gives the exact count of primes for every finite x.

Watch the explanation

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