π(n) counts the primes up to n (blue staircase). The Prime Number Theorem says π(n) ~ n/ln(n) — the ratio → 1 as n → ∞. The logarithmic integral Li(n) is even closer.
Table showing density of primes at various scales
| Bis n | Primzahlen π(n) | Dichte ≈ 1/ln(n) |
|---|---|---|
| 1 000 | 168 | 1 von 7 |
| 1 000 000 | 78 498 | 1 von 14 |
| 10⁹ | 50 847 534 | 1 von 21 |
| 10¹² | 37 607 912 018 | 1 von 28 |