"The definitions of and are more complicated than that of , but they are in reality more 'natural' functions. Thus is [by definition] the 'sum function' of , and has a simple generating function. The generating functions of and still more of , are much more complicated. And even the arithmetical definition of , written in the form [ = log U(x)], is very elementary and natural." (Here U(x) is the least common multiple of {1,2,3,...,[x].)

[Hardy and Wright, An Introduction to the Theory of Numbers]

"It happens that, of the three functions , , , the one which arises most naturally from the analytical point of view is the one most remote from the original problem, namely . For this reason, it is usually most convenient to work in the first instance with ...to deduce results about . This is a complication which seems inherent in the subject, and the reader should familiarise himself at the outset with the function , which is to be regarded as the fundamental one."

[A.E. Ingham, The Distribution of Prime Numbers, p.13]

"[medaevil philosopher] William of Occam, elevated to a method the idea that when one must choose between two explanations, one should always choose the simpler. Occam's razor, as the principle is called, cuts out the difficult and chooses the simple. When things get too complicated, it sometimes makes sense to stop and wonder: Have I asked the right question? Here the choice is between two functions that count primes: one is the function psi(x), approximated by a straight line, and the second is pi(x), approximated by the curving function Li(x). Surely William of Occam would have chosen to study psi(x)."

[E. Bombieri from "Prime Territory: Exploring the Infinite Landscape at the Base of the Number System" (The Sciences, Sept/Oct 1992)]

The other Chebyshev Function counts all primes p less than or equal to x with value log p. is related as follows:

= (x) + (x^1/2) + (x^1/3) + . . .

Note that the right-hand side will only have a finite number of nonzero terms, as there will be no primes less than x^1/n for sufficiently large n.

Both and are asymptotic to x, and these assertions are both equivalent to the PNT.

In Introduction to Analytic Number Theory, T. Apostol explains how the von Mangolt function arises naturally from the Fundamental Theorem of Arithmetic, by proving that log n is the sum of the von Mangoldt values of the divisors of n, that is

Although counts all powers of the prime p with the same value, log p, the fact that log(p^k) = klog(p) suggests that there is a sort of "inverse-power weighting" involved here. That is, the power p^k is being counted with value log(p^k)/k.

This might make more sense if you consider the following:

The approximation of given by x/log x can be improved by Li(x), its integral equivalent.

We can further go on to observe that Li(x) gives a much better fit still if we also count nth powers of primes with the value 1/n.

Thus Li(x) is best thought of as an approximation of , a function which counts all powers of primes with "inverse-power weight".