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September 2007 Cramér vs. Cramér. On Cramér's probabilistic model for primes
János Pintz
Funct. Approx. Comment. Math. 37(2): 361-376 (September 2007). DOI: 10.7169/facm/1229619660


In the 1930's Cramér created a probabilistic model for primes. He applied his model to express a very deep conjecture about large differences between consecutive primes. The general belief was for a period of 50 years that the model reflects the true behaviour of primes when applied to proper problems. It was a great surprise therefore when Helmut Maier discovered in 1985 that the model gives wrong predictions for the distribution of primes in short intervals. In the paper we analyse this phenomen, and describe a simpler proof of Maier's theorem which uses only tools available at the mid thirties. We present further a completely different contradiction between the model and the reality. Additionally, we show that, unlike to the contradiction discovered by Maier, this new contradiction would be present in essentially all Cramér type models using independent random variables.


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János Pintz. "Cramér vs. Cramér. On Cramér's probabilistic model for primes." Funct. Approx. Comment. Math. 37 (2) 361 - 376, September 2007.


Published: September 2007
First available in Project Euclid: 18 December 2008

zbMATH: 1226.11096
MathSciNet: MR2363833
Digital Object Identifier: 10.7169/facm/1229619660

Primary: 11N05

Keywords: Cramér's model for primes , primes , probabilistic model for primes

Rights: Copyright © 2007 Adam Mickiewicz University


Vol.37 • No. 2 • September 2007
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