e as a log.........................ln...................tends to go under the number..........pi as a log.............tends to go over them, overestimate them............................................pi is closer, at least at 1st............

Below is about ln..........using e as a log......................


This estimate systematically underestimates the numbers of primes, except possibly for staggeringly large values of n$$nn . As a theorem it would state that in the limit of n$$nn going to infinity, the ratio of π(n)$$π⁢n π n to nln(n)$$nlnn n n 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)$$$$Li⁢n Li n

where Li$$LiLi , the log integral function is defined to be

∫n21ln(t)dt$$$$∫2n1lntdt t 2 n 1 t