the prime number theorem
"Some order begins to emerge from this chaos when the primes are considered not
in their individuality but in the aggregate; one considers the social statistics of the
primes and not the eccentricities of the individuals."
P.J. Davis and R. Hersh,
The Mathematical Experience, Chapter 5
hyperlinked proof outline
Wikipedia: prime number theorem
WolframMathworld: prime number theorem
Ilan Vardi, "Introduction to Analytic Number Theory" including useful notes on the PNT
Chapter 5 of P.J. Davis and R. Hersh's The
Mathematical Experience, dealing with the PNT
a proof of the prime number theorem involving Fourier transforms (summarised by Jonas Wiklund)
Although not its primary subject matter,
this article by A. Granville contains a beautifully succinct explanation of the PNT, in a historical context.
"The prime number theorem obtained by statistical methods" – a heuristic argument from What is Mathematics? by Courant and Robbins
A. Selberg, "An elementary proof of the prime-number theorem", Annals of Mathematics 50 No. 2 (1949) 305–313
A new way to visualise the Prime Number Theorem – approximate logarithmic spirals generated from the distribution of primes
P.T. Bateman, "Major figures in the history of the prime number theorem", Abstracts of the American Mathematical Society (87th annual meeting, San Francisco), 1981, p.2.
N. Wiener's The Fourier Integral and Certain of its Applications, section 17,
"The Prime-Number Theorem as a Tauberian Theorem" begins: "The present section and the three following will be devoted to the application of Tauberian theorems to the problem of the distribution of the primes. The theorem which we
shall eventually prove is the famous theorem of Hadamard and de la Vallée Poussin..."
number theory and physics archive