Logarithmic integral

The logarithmic integral, commonly written $\operatorname{Li}(x)$, is a function that gives a remarkably good approximation to the number of primes up to $x$.

Riemann Console uses it in the classic warning that long-lived numerical patterns need not hold forever: the difference between the actual prime count $\pi(x)$ and $\operatorname{Li}(x)$ eventually changes sign, even though one ordering persists across an enormous range.

See also: prime-counting function, Prime Number Theorem, numerical evidence, sign..