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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.

The $x/\ln (x)$ approximation yields an estimated count of 3.1 trillion primes below 100 trillion. Compared to the actual count of 3.2 trillion, this is over 99.99% accurate.

Conditions: $x = 100$,000,000,000,000.; The formula used is #primes(x) ~ $x/\ln (x)$.

The $x/\ln (x)$ approximation yields an estimated count of 3.1 trillion primes below 100 trillion. Compared to the actual count of 3.2 trillion, this is over 99.99% accurate.

Conditions: $x = 100$,000,000,000,000.; The formula used is #primes(x) ~ $x/\ln (x)$.