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The video derives the Prime Number Theorem by treating the number of primes up to x as the area under the density curve. It simplifies this by assuming constant density, multiplying the size x by the density estimate $1/\ln (x)$ to get $x/\ln (x)$.

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

The video derives the Prime Number Theorem by treating the number of primes up to x as the area under the density curve. It simplifies this by assuming constant density, multiplying the size x by the density estimate $1/\ln (x)$ to get $x/\ln (x)$.

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