Maier's theorem

In number theory, Maier's theorem (Maier 1985) is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives the wrong answer.

The theorem states that if π is the prime counting function and λ is greater than 1 then

{\frac {\pi (x+(\log x)^{\lambda })-\pi (x)}{(\log x)^{\lambda -1}}}

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).

Proofs

Maier proved his theorem using Buchstab's equivalent for the counting function of quasi-primes (set of numbers without prime factors lower to bound $z=x^{1/u}$ , $u$ fixed). He also used 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

\int _{2}^{Y}\left(\sum _{2<p\leq x}\log p-\sum _{2<n\leq x}1\right)^{2}\,dx

of one version of the prime number theorem.

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References

Maier, Helmut (1985), "Primes in short intervals", The Michigan Mathematical Journal, 32 (2): 221–225, doi:10.1307/mmj/1029003189, ISSN 0026-2285, MR 0783576, 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 (eds.), 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

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