Maier's theorem, proved by Helmut Maier and published in 1985 in the Michigan Mathematical Journal, exposes a flaw in Cramer's probabilistic model of the distribution of primes. It shows that for the count of primes in short intervals of a certain length, the limit superior of the relevant ratio exceeds 1 and the limit inferior falls below 1, contradicting Cramer's prediction that the ratio should converge to exactly 1. The result demonstrated that probabilistic heuristics, while useful, do not fully capture the real behavior of primes in short intervals. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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StatementThe ratio (pi(x + (log x)^lambda) - pi(x)) / (log x)^(lambda - 1), for lambda > 1, has no limit as x tends to infinity: its limit superior is greater than 1 and its limit inferior is less than 1. 1 Classification
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Source Maier's theorem (Wikipedia)
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1. Maier's theorem (Wikipedia)
Introduction, second paragraph
the limit superior is greater than 1, and the limit inferior is less than 1
In Branch: Number Theory, Lead sentence
In number theory, Maier's theorem is a theorem due to Helmut Maier about the numbers of primes in short intervals for which Cramér
Proved By: Helmut Maier, Lead
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.
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