Maier's theorem
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 states that if π is the prime-counting function and λ > 1, then
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 λ ≥ 2.
Proofs
Maier proved his theorem using Buchstab's equivalent for the counting function of quasi-primes. He also used an equivalent of the number of primes in arithmetic progressions of sufficient length due to Gallagher.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.