1/ln x curve
The Prime Number Theorem says primes are distributed with density 1/ln x.
So the area under 1/ln x between two numbers should be close to the
actual number of primes between them. Pick a range and watch it.
| x | π(x) — real primes | Li(x) = ∫ 1/ln t dt | Li(x) − π(x) |
|---|
Legendre's guess x/ln x was only an approximation; the better estimate is the
logarithmic integral Li(x) = ∫₀ˣ dt/ln t. The area between two numbers — not the
height of the curve — is what counts primes. (Actual π(x) is computed by a sieve in your browser.)