Functiones et Approximatio Commentarii Mathematici

Cramér vs. Cramér. On Cramér's probabilistic model for primes

János Pintz

Full-text: Open access

Abstract

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.

Article information

Source
Funct. Approx. Comment. Math. Volume 37, Number 2 (2007), 361-376.

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.facm/1229619660

Digital Object Identifier
doi:10.7169/facm/1229619660

Subjects
Primary: 11N05: Distribution of primes

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

Citation

Pintz, János. Cramér vs. Cramér. On Cramér's probabilistic model for primes. Functiones et Approximatio Commentarii Mathematici 37 (2007), no. 2, 361--376. doi:10.7169/facm/1229619660. http://projecteuclid.org/euclid.facm/1229619660.


Export citation

References

  • A.A. Buchstab, Asymptotic estimates of a general number-theoretic function (Russian), Mat. Sbornik (N.S.) 2 (44) (1937), 1239--1246.
  • H. Cramér, Some theorems concerning prime numbers, Arkiv f. Math. Astr. Fys. 15, No. 5 (1920), 1--33.
  • H. Cramér, Prime numbers and probability, Skand. Math. Kongr. 8 (1935), 107--115.
  • H. Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 23--46.
  • H. Davenport, Multiplicative Number Theory, Revised by Hugh L. Montgomery, 2$^\hbox\rrmm nd$ edition, Springer, Berlin, Heidelberg, New York, 1980.
  • P.X. Gallagher, A large sieve density estimate near $\sigma = 1$, Invent Math. 11 (1970), 329--339.
  • A. Granville, Harald Cramér and the Distribution of Prime Numbers, Scand. Actuarial J. 1995, No. 1, 12--28.
  • A. Granville, Unexpected irregularities in the distribution of prime numbers, in: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 388--399, Birkhäuser, Basel, 1995.
  • H. Iwaniec, The sieve of Eratosthenes--Legendre, Ann Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 257--268.
  • H. von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1901), 159--182.
  • J.E. Littlewood, Sur la distribution des nombres premiers, Comptes Rendus Acad. Sci. Paris 158 (1914), 1869--1872.
  • H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985), 221--225.
  • E. Phragmén, Sur le logarithme intégral et la fonction $f(x)$ de Riemann, Ofversight Kongl. Vet.-Akad. Förh. Stockholm 48 (1891), 599--616.
  • J. Pintz, On the remainder term of the prime number formula I. On a problem of Littlewood, Acta Arith. 36 (1980), 341--365.
  • G. Pólya, Heuristic reasoning in the theory of numbers, Amer. Math. Monthly 66 (1959), 375--384.
  • Sz.Gy. Révész, Effective oscillation theorems for a general class of real-valued remainder terms, Acta Arith. 49 (1988), 481--505.
  • C.L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935), 83--86.
  • P. Turán, On the remainder term of the prime number formula I, Acta Math. Hungar. 1 (1950), 48--63.