Let \(\pi(x)\) denote the prime-counting function, i.e. the number of primes \(p \le x\).
A classical first-order approximation is
\[ \pi(x)\;\approx\;\frac{x}{\ln x}. \]
Sarko’s Prime Counting Function is the explicit approximation \(S(x)\) defined by
\[ S(x)\;=\;\frac{x\Bigl(1-\sqrt{\,1-\frac{4}{\ln(x)}\,}\Bigr)}{2}. \]
The goal is to compare \(S(x)\) against \(\frac{x}{\ln x}\) on standard benchmarks \(x=10^{10},10^{12},10^{14},10^{16}\).
The table below uses the known exact values of \(\pi(10^n)\) and reports each approximation’s absolute and relative error.
| \(x\) | \(\pi(x)\) | \(\frac{x}{\ln x}\) | \(S(x)\) | \(\frac{x}{\ln x}-\pi(x)\) | \(S(x)-\pi(x)\) | rel. err \(\frac{x}{\ln x}\) | rel. err \(S(x)\) |
|---|---|---|---|---|---|---|---|
| \(10^{10}\) | 455,052,511 | 434,294,482 | 454,996,680 | −20,758,029 | −55,831 | −4.562% | −0.012269% |
| \(10^{12}\) | 37,607,912,018 | 36,191,206,825 | 37,605,370,733 | −1,416,705,193 | −2,541,285 | −3.767% | −0.006757% |
| \(10^{14}\) | 3,204,941,750,802 | 3,102,103,442,166 | 3,204,811,617,182 | −102,838,308,636 | −130,133,620 | −3.209% | −0.004060% |
| \(10^{16}\) | 279,238,341,033,925 | 271,434,051,189,532 | 279,231,049,065,770 | −7,804,289,844,393 | −7,291,968,155 | −2.795% | −0.002611% |
Across these benchmarks, \(\tfrac{x}{\ln x}\) undershoots \(\pi(x)\) by roughly \(2.8\%\)–\(4.6\%\), while \(S(x)\) stays within about \(0.0026\%\)–\(0.012\%\) (still a slight undershoot in each case). In absolute terms, \(S(x)\) reduces the error by orders of magnitude relative to \(\tfrac{x}{\ln x}\) at each tested scale.