Prime counting approximations: π(x), Li(x), and R(x)¶
The prime-counting function (\pi(x)) counts primes up to (x). Two standard smooth approximations are:
the logarithmic integral (\mathrm{Li}(x)), and
Riemann’s R function (R(x)) (a Möbius-weighted combination of (\mathrm{Li}(x^{1/n}))).
These approximations are central for numerical experiments around the Prime Number Theorem, error terms, and the visual “shape” of prime races.
Key ideas¶
PNT baseline: (\pi(x)\sim \frac{x}{\log x}), and (\mathrm{Li}(x)) is often an excellent smooth baseline.
R(x) as a refinement: R(x) folds in prime-power information and is closely connected to explicit-formula viewpoints.
Error terms: plotting (\pi(x)-\mathrm{Li}(x)) or (\psi(x)-x) reveals oscillations and sign changes.
References¶
See References.