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Cramér vs. Cramér. On Cramér’s probabilistic model for primes. (English) Zbl 1226.11096
Summary: In the 1930’s H. Cramér [8. Skand. Mat.-Kongr. 1934, 107–115 (1935; Zbl 0011.40801 and JFM 61.1051.01)] 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 H. Maier discovered in 1985 [Mich. Math. J. 32, 221–225 (1985; Zbl 0569.10023)] that the model gives wrong predictions for the distribution of primes in short intervals. In the paper we analyse this phenomenon, 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.
MSC:
11N05Distribution of primes