Maier's theorem
In number theory, Maier's theorem is a theorem 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 is greater than 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 (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 formula_2, formula_3 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.