In mathematics, Maier's theorem is a theorem about the numbers of primes in short intervals for which many probabilistic models of primes give the wrong answer. It states that if π is the prime counting function and λ is greater than 1 then <math>\frac{\pi(x+^\lambda)-\pi(x)}{(\log x)^{\lambda-1}}</math> 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.

PropertyValue
dbpprop:abstract
  • In mathematics, Maier's theorem is a theorem about the numbers of primes in short intervals for which many probabilistic models of primes give the wrong answer. It states that if π is the prime counting function and λ is greater than 1 then <math>\frac{\pi(x+^\lambda)-\pi(x)}{(\log x)^{\lambda-1}}</math> 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. Many probabilistic models of primes predict incorrectly that it has limit 1 when λ≥4. Pintz (2007) gave another proof, and also showed that most probabilistic models of primes incorrectly predict the mean square error <math>\int_2^Y\left(\sum_{2<p\le x} \log p -\sum_{2<n\le x}1\right)^2\,dx</math> of one version of the prime number theorem.
dbpprop:harvtxtProperty
  • Pintz
  • 2007 (xsd:integer)
dbpprop:relatedInstance
dbpprop:wikiPageUsesTemplate
rdfs:comment
  • In mathematics, Maier's theorem is a theorem about the numbers of primes in short intervals for which many probabilistic models of primes give the wrong answer. It states that if π is the prime counting function and λ is greater than 1 then <math>\frac{\pi(x+^\lambda)-\pi(x)}{(\log x)^{\lambda-1}}</math> 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.
rdfs:label
  • Maier's theorem
skos:subject
foaf:page