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

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

#### 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.

First Page:
Primary Subjects: 11N05
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.facm/1229619660
Digital Object Identifier: doi:10.7169/facm/1229619660

### 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.
Mathematical Reviews (MathSciNet): MR606931
P.X. Gallagher, A large sieve density estimate near $\sigma = 1$, Invent Math. 11 (1970), 329--339.
Mathematical Reviews (MathSciNet): MR279049
Zentralblatt MATH: 0219.10048
Digital Object Identifier: doi:10.1007/BF01403187
A. Granville, Harald Cramér and the Distribution of Prime Numbers, Scand. Actuarial J. 1995, No. 1, 12--28.
Mathematical Reviews (MathSciNet): MR1349149
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.
Mathematical Reviews (MathSciNet): MR1403939
Zentralblatt MATH: 0843.11043
H. Iwaniec, The sieve of Eratosthenes--Legendre, Ann Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 257--268.
Mathematical Reviews (MathSciNet): MR453676
H. von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1901), 159--182.
Mathematical Reviews (MathSciNet): MR1554926
Digital Object Identifier: doi:10.1007/BF02403071
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.
Mathematical Reviews (MathSciNet): MR783576
Digital Object Identifier: doi:10.1307/mmj/1029003189
Project Euclid: euclid.mmj/1029003189
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.
Mathematical Reviews (MathSciNet): MR585891
G. Pólya, Heuristic reasoning in the theory of numbers, Amer. Math. Monthly 66 (1959), 375--384.
Mathematical Reviews (MathSciNet): MR104639
Digital Object Identifier: doi:10.2307/2308748
Sz.Gy. Révész, Effective oscillation theorems for a general class of real-valued remainder terms, Acta Arith. 49 (1988), 481--505.
Mathematical Reviews (MathSciNet): MR967333
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.
Mathematical Reviews (MathSciNet): MR43121
Zentralblatt MATH: 0040.01601
Digital Object Identifier: doi:10.1007/BF02022552