3.4 $\pi[x]$: Base Prime Counting Function

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$\pi[x]$ – Base Prime Counting Function that takes a unit step at each prime integer.

The following $\pi[x]$ function is the base prime-number counting function which takes a unit step at each prime.

$\quad\pi[x]=\sum_{i=1}^{\lfloor x\rfloor}If[PrimeQ[i], 1, 0]$

The following plot illustrates the $\pi[x]$ function (blue), the $x\ /\log[x]$ function (green), and the RiemannR[x] function (orange). The $x\ /\log[x]$ function is a historical estimate for $\pi[x]$, whereas the RiemannR[x] function (defined on page 3.2) is a more modern and accurate estimate for $\pi[x]$.

Base Prime Counting Function (p[x])
Base Prime Counting Function $\pi[x]$