The theorem states that if π is the prime counting function and λ is greater than 1 then
does not have a limit as x tends to infinity; more precisely the lim sup is greater than 1, and the lim inf is less than 1. The Cramér model of primes predicts incorrectly that it has limit 1 when λ≥2 (using the Borel–Cantelli lemma).
Maier proved his theorem using Buchstab's equivalent for the counting function of quasi-primes (set of numbers without prime factors lower to bound , fixed). He also used an equivalent of the number of primes in arithmetic progressions of sufficient length due to Gallagher.
of one version of the prime number theorem.
- Maier, Helmut (1985), "Primes in short intervals", The Michigan Mathematical Journal, 32 (2): 221–225, doi:10.1307/mmj/1029003189, ISSN 0026-2285, MR 783576, Zbl 0569.10023
- Pintz, János (2007), "Cramér vs. Cramér. On Cramér's probabilistic model for primes", Functiones et Approximatio Commentarii Mathematici, 37: 361–376, doi:10.7169/facm/1229619660, ISSN 0208-6573, MR 2363833, Zbl 1226.11096
- Soundararajan, K. (2007), "The distribution of prime numbers", in Granville, Andrew; Rudnick, Zeév, Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study Institute on equidistribution in number theory, Montréal, Canada, July 11--22, 2005, NATO Science Series II: Mathematics, Physics and Chemistry, 237, Dordrecht: Springer-Verlag, pp. 59–83, ISBN 978-1-4020-5403-7, Zbl 1141.11043