I do not think it is based off of one number..........or process.........or even one way that number or process is expressed.............it is cyclical.......







The PNT as a theorem asserts that the following interpolation formulas, which attempt to smoothly fit onto the trends in the distribution of the primes, are asymptotically exact.

An early estimate was that π(n)$\pi \left(n\right)$ was approximately

nln(n)$$\frac{n}{\ln \left(n\right)}$$

This estimate systematically underestimates the numbers of primes, except possibly for staggeringly large values of n$n$ . As a theorem it would state that in the limit of n$n$ going to infinity, the ratio of π(n)$\pi \left(n\right)$ to nln(n)$\frac{n}{\ln \left(n\right)}$ is unity. The theorem is of particular interest to mathematicians in that it follows from an as-yet unproven conjecture by Riemann on the distribution of zeros of the zeta function, a complicated topic which is not required for our simple analysis here.

An improved approximation is

Li(n)$$\mathrm{Li}\left(n\right)$$

where Li$\mathrm{Li}$ , the log integral function is defined to be

∫n21ln(t)dt

This estimate is substantially better and systematically underestimates the number of primes (again until one gets to staggeringly large values). Notice that the first approximation results from considering the log term to be slowly varying and taking it out of the integral to get the (crude) estimate of nln(n)$\frac{n}{\ln \left(n\right)}$ . The log