- Maier's theorem
-
In number theory, Maier's theorem (Maier 1985) is a theorem about the numbers of primes in short intervals for which the Cramér's probabilistic model of primes gives the wrong answer.
It 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's probabilistic model of primes predicts incorrectly that it has limit 1 when λ≥2 (using the Borel–Cantelli lemma).
Maier's theorem uses the beautiful Buchstab's equivalent for the counting function of quasi-primes (set of numbers without prime factors lower to bound z = x1 / u, u fixed). It also uses an equivalent of the number of primes in arithmetic progressions of sufficient length due to Gallagher.
Pintz (2007) gave another proof, and also showed that most probabilistic models of primes incorrectly predict the mean square error
of one version of the prime number theorem.
References
- Maier, Helmut (1985), "Primes in short intervals", The Michigan Mathematical Journal 32 (2): 221–225, doi:10.1307/mmj/1029003189, ISSN 0026-2285, MR783576, http://projecteuclid.org/euclid.mmj/1029003189
- Pintz, János (2007), "Cramér vs. Cramér. On Cramér's probabilistic model for primes", Functiones et Approximatio Commentarii Mathematici 37: 361–376, MR2363833, http://projecteuclid.org/euclid.facm/1229619660
This number theory-related article is a stub. You can help Wikipedia by expanding it.