« on: Yesterday at 20:53:59 »
Quote from: evan_au
Perhaps more mind-bending, they also say that the maximum Absolute Error of Li(x)=∫1/ln(t) grows without limit as x→∞.Thinking about this some more... (and viewing the graph)...
When searching through large numbers, primes occur at (pseudo-)random intervals.
- While Li(x) is a long-term average of the expected number of primes.
- So the Absolute Error |Li(x) - π(x)| is actually a (pseudo-)random walk, which can stray large distances from 0
- If it were a true random walk, the standard deviation of absolute error would grow as √x.
- And maximum Absolute Error |Li(x) - π(x)| → ∞ as x → ∞
But this is still compatible with π(x)/Li(x) → 1 as x → ∞
..since Relative Error (√x)/x → 0 as x → ∞