Skip to main content

All Questions

Filter by
Sorted by
Tagged with
32 votes
3 answers
8k views

Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)

I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949). One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$, $$(1) \qquad\qquad \vartheta(x) \...
Basj's user avatar
  • 587
21 votes
4 answers
4k views

What is the difference between elementary and non-elementary proofs of the Prime Number Theorem?

There is an easy proof of the PNT, just in a few lines, in the book by Julian Havil, "Gamma", pages 201-202. Specifically, Von Mangoldt's formula, which is very easy to derive: $$ \psi(x) = ...
Peter S.'s user avatar
  • 221
7 votes
2 answers
636 views

How to use the Prime Number Theorem in order to prove Selberg's Formula?

I`m reading Melvin B. Nathanson's "Elementary Methods in Number Theory" and I can't think of a way of deducing Selberg's formula (9.3) from the prime number theorem. This is one of the tasks ...
Juu's user avatar
  • 129