zbMATH — the first resource for mathematics

A fibered system associated with the prime number sequence. (Sur un système fibré lié à la suite des nombres premiers.) (French) Zbl 1117.37301
Summary: We study the dynamical system defined by the transformation \(\Phi:\,]0,1]\leq ]0,1]\) where \(\Phi(x)=px-1\) if \(x\in ]1/p,1/q],\) \(q\) and \(p\) being two consecutive prime numbers. The problem of the existence of an invariant absolutely continuous measure by \(\Phi\) is related via a Markov chain argument to a conjecture concerning a set of prime number sequences. This hypothesis is corroborated by Monte Carlo simulations. We prove that this implies the statistical stability of the transformation \(\Phi\) on the interval \(]0,2/3].\) By using heuristic arguments, we define simplified versions of the Perron-Frobenius operator associated to \(\Phi.\) Using Maple, we construct a probability density presenting a good experimental fit with the histograms of orbits stemming from fundamental constants.
37E05 Dynamical systems involving maps of the interval
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
Full Text: DOI Euclid EuDML
[1] Cramér H., Acta. Arith. 2 pp 396– (1936)
[2] Ellison W. J., Les nombres premiers (1975) · Zbl 0313.10001
[3] Feller W., An introduction to Probability Theory and Its Applications (1968) · Zbl 0155.23101
[4] Lasota A., Chaos, Fractals, and Noise (1994)
[5] Schweiger F., Ergodic theory of fibered systems and metric number theory (1995) · Zbl 0819.11027
[6] Ulam S. M., A collection of Mathematical problems 8 (1960)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.