Maier's theorem

Template:Short description In number theory, Maier's theorem is a theorem due to Helmut Maier about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer.

The theorem statesTemplate:Sfnp that if $\pi$ is the prime-counting function and $\lambda >1$, 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 limit superior is greater than 1, and the limit inferior is less than 1. The Cramér model of primes predicts incorrectly that it has limit 1 when $\lambda \ge 2$ (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 $z=x^{1/u}$, with $u$ fixed). He also used an equivalent of the number of primes in arithmetic progressions of sufficient length due to Gallagher.

János Pintz gave another proof,Template:Sfnp and also showed that most probabilistic models of primes incorrectly predict the mean square error

${\int }_2^Y(\sum _{2<p\le x}\log p-\sum _{2<n\le x}1{)}^{2}dx$

of one version of the prime number theorem.

See also

• Maier's matrix method

References

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