The So-called Renormalization Group Method Applied to the Specific Prime Numbers Logarithmic Decrease

A so-called Renormalization Group (RG) analysis is performed in order to shed some light on why the density of prime numbers in N decreases like the single power of the inverse neperian logarithm. CERN–TH/2000–146 May 2000 Part I The most elementary proof of the Prime Numbers Theorem These few lines are not part of the proof. They simply show the history which has led to the starting point of our proof. The two main steps of it involve only formal elementary algebra, with no recourse naturally to Functions’ theory nor complex variables. Euler proved in 1747 [1], quite formally, that the prime numbers are linked with natural integers, by establishing what is universally known as “Euler identity” and which is so famous that we do not recall it here. From this identity, one can deduce straight forwardly an approximate formula giving (I) ∑ p<Λ 1/p = 1 · log(log Λ) + · · · . This result has been refined, mainly by Mertens [2] with the aim to establish the value of constants possibly entering a much more exact expression of the sum in (I). This expression, we call it “Euler-Mertens identity” and is the starting point of our proof, the formula 1 of our theorem.