The Prime Number Theorem: the “elementary” proof

The goal

The goal of this path is to prove one of the most important theorems about prime numbers, the so called Prime Number Theorem, which can be expressed by the following formula:

\pi(x) \sim \frac{x}{\log x}

where \pi(x) indicates the number of prime numbers less than or equal to x. The symbol \sim represents an asymptotic equivalence, thw meaning of which is explained in one of our posts. However, to make things simple, we can read the formula saying that there are about \frac{x}{\log x} prime numbers less than or equal to x, and this estimate becomes more and more accurate as x increases. We encourage you to verify it with different values of x.

We’ll illustrate the proof by Paul Erdős (alongside in the image) and Atle Selberg. When it was proposed for the first time in 1949, the mathematical community felt astonished, because it’s based exclusively on real analysis, differently from the previously known proof, which instead uses sophisticated notions of complex analysis. For this reason Erdős and Selberg’s proof is called “elementary”, in spite of its considerable complexity: the term “elementary” in fact is referred exclusively to the employed techniques, rather than to the proof itself.

The path

Along the path towards the proof of the Prime Number Theorem, we’ll introduce gradually several notions of number theory and we’ll prove several intermediate results of some importance, as we’ll move close to the final result. The most important intermediate result will be the strong version of Chebyshev’s Theorem, which is the goal of another path, which this one is the continuation of.
Generally, we’ll attempt to adopt an approach which is similar to the one of detective stories, in which several evidences are collected and, thanks to them, the various aspects of the story become more and more clear, up to identifying the culprit. In the same way, we’ll attempt to move close to the final result by studying in detail different themes separately, up to obtain the overall picture.

Map of paths for number theory
General introduction to paths