Prime Number Theorem - Proof Sketch

Proof Sketch

Here is a sketch of the proof referred to in Tao's lecture mentioned above. Like most proofs of the PNT, it starts out by reformulating the problem in terms of a less intuitive, but better-behaved, prime-counting function. 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

Here the summation is over all prime powers up to x. 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 O notation) for any ε > 0,

The next step is to find a useful representation for . Let be the Riemann zeta function. It can be shown that is related to the von Mangoldt function, and hence to, via the relation

A delicate analysis of this equation and related properties of the zeta function, using the Mellin transform and Perron's formula, shows that for non-integer x the equation

holds, where the sum is over all zeros (trivial and non-trivial) of the zeta function. This striking formula is one of the so-called explicit formulas of number theory, and is already suggestive of the result we wish to prove, since the term x (claimed to be the correct asymptotic order of ) appears on the right-hand side, followed by (presumably) lower-order asymptotic terms.

The next step in the proof involves a study of the zeros of the zeta function. The trivial zeros −2, −4, −6, −8, ... can be handled separately:

which vanishes for a large x. The nontrivial zeros, namely those on the critical strip, can potentially be of an asymptotic order comparable to the main term x if, so we need to show that all zeros have real part strictly less than 1.

To do this, we take for granted that is meromorphic in the half-plane, and is analytic there except for a simple pole at, and that there is a product formula for This product formula follows from the existence of unique prime factorization of integers, and shows that is never zero in this region, so that its logarithm is defined there and Write ; then

Now observe the identity so that

for all . Suppose now that . Certainly is not zero, since has a simple pole at . Suppose that and let tend to from above. Since has a simple pole at and stays analytic, the left hand side in the previous inequality tends to, a contradiction.

Finally, we can conclude that the PNT is "morally" true. To rigorously complete the proof there are still serious technicalities to overcome, due to the fact that the summation over zeta zeros in the explicit formula for does not converge absolutely but only conditionally and in a "principal value" sense. There are several ways around this problem but all of them require rather delicate complex-analytic estimates that are beyond the scope of this article. Edwards's book provides the details.