Elementary proof of the Prime Number Theorem
Statement
The Prime Number Theorem π(x) ~ x / ln x as x → ∞ can be proved without complex analysis or the Riemann zeta function on the critical line. Selberg and Erdős, in 1949, gave such elementary arguments (building on real-variable estimates, including Selberg’s symmetric function and his “elementary” sieve inequality). This complemented the classical analytic proofs of Hadamard and de la Vallée Poussin (1896). The result is a theorem of ordinary mathematics, formalizable in ZFC.
Formula
π(x) ~ x / ln x (elementary methods)
Equation
lim_{x→∞} π(x) / (x / ln x) = 1
Proof sketch
Selberg’s Annals paper (1949) gives a self-contained elementary derivation. Erdős’s PNAS note (1949) obtains the same asymptotic using Selberg’s inequality. No use of ζ(s) in the complex plane in the core argument.
Source
Selberg, A. An elementary proof of the prime-number theorem. Ann. of Math. (2) 50 (1949), 305–313. Erdős, P. On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Natl. Acad. Sci. USA 35 (1949), 374–384.
Domains
mathematics
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- Proof: Elementary proof of the Prime Number Theorem