Math 609, Prime Number Theorem: Some ReferencesA Google search using "prime number theorem" gives many useful items, including some of those listed here.
Riemann's brilliant 1859 article outlines the now classical approach.
The current simplest proof of the prime number theorem using complex analysis follows the 1980 approach of D.J. Newman, usually in J. Korevaar's 1982 version - as in many of the items below (and also Lang's "Complex Analysis" text). Korevaar's 2002 article gives some recent improvement and insight on the analytic aspects.
- How many primes are there?
- Prime Number Theorem -- from MathWorld
- Prime Number Theorem Lecture Notes (Matt Baker and Dennis Clark)
- Zeta(z): No zeroes on x=1 (Math 609 Lecture Notes)
- P. Bateman and H. Diamond, "A hundred years of prime numbers," Amer. Math.Monthly 103 (1996), pp. 729--741.
- J. Korevaar, "On Newman's quick way to the prime number theorem," Math. Intelligencer 4 (1982), pp. 108--115.
- J. Korevaar, "A century of complex Tauberian Theorey," Bull. Amer. Math. Soc., 39 (2002), pp. 475--531.
- D. J. Newman, "Simple Analytic proof of the prime number theorem," Amer. Math.Monthly 87 (1980), pp. 693-696.
- Bernhard Riemann, "Uber die Anzahl der Primzahlen unter einer gegebenen Grosse," Monatsberichte der Berliner Akademie, Nov. 1859. ( Original version and English translation )
- D. Zagier, "Newman's short proof of the prime number theorem," Amer. Math. Monthly, 104 (1997), pp. 705--708