A new elementary proof of the Prime Number Theorem

The Elementary Proof of the Prime Number Theorem

P rime numbers are the atoms of our mathematical universe. Euclid showed that there are infinitely many primes, but the subtleties of their distribution continue to fascinate mathematicians. Letting p(n) denote the number of primes p B n, Gauss conjectured in the early nineteenth century that pðnÞ#n=lnðnÞ. In 1896, this conjecture was proven independently by Jacques Hadamard and Charles de la V...

On a New Method in Elementary Number Theory Which Leads to An Elementary Proof of the Prime Number Theorem.

Simple Proof of the Prime Number Theorem

A form of this was conjectured by Gauss about 1800, [Chebyshev 1848/52] and [Chebyshev 1850/52] made notable progress with essentially elementary methods. The landmark paper Riemann 1859] made clear the intimate connection between prime numbers and the behavior of ζ(s) as a function of a complex variable. The theorem was proven independently by [Hadamard 1896] and [de la Vallée Poussin 1896] by...

A formally verified proof of the prime number theorem (draft)

The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdös in 1948. We describe a formally verified version of Selberg’s proof, obtained using th...

Simple Proof of the Prime Number Theorem , etc

The point here is the relatively simple argument that non-vanishing of an L-function on the line Re (s) = 1 implies an asymptotic result parallel to the application of ζ(s) to the Prime Number Theorem. This is based upon [Newman 1980]. In particular, this argument avoids estimates on the zeta function at infinity and also avoids Tauberian arguments. For completeness, we recall the standard clev...