 The idea is to count the primes (or a related set such as the set of prime powers) with "weights" to arrive at a function with smoother asymptotic behavior.The most common such generalized counting function is the Chebyshev function , defined by

This is sometimes written as

where is the von Mangoldt function, namely

It is now relatively easy to check that the PNT is equivalent to the claim that
Indeed, this follows from the easy estimates
and (using big notation) for any ,

The next step is to find a useful representation for .Let be the Riemann zeta function.