IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i11p1224-d563588.html
   My bibliography  Save this article

Combinatorial Models of the Distribution of Prime Numbers

Author

Listed:
  • Vito Barbarani

    (European Physical Society, via Cancherini 85, 51039 Quarrata, Italy)

Abstract

This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n -th prime number is derived together with an estimate of the prime-counting function π ( x ) . A proposition equivalent to the Prime Number Theorem ( PNT ) is proved to hold, while the equivalent of the Riemann Hypothesis ( RH ) is proved to be false with probability 1 ( w.p. 1 ) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH . A model representing a general class of random integer sequences is found, for which RH holds w.p. 1 . The prediction of the number of consecutive prime pairs as a function of the gap d , is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π ( x ) are found.

Suggested Citation

  • Vito Barbarani, 2021. "Combinatorial Models of the Distribution of Prime Numbers," Mathematics, MDPI, vol. 9(11), pages 1-50, May.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:11:p:1224-:d:563588
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/11/1224/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/11/1224/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Wolf, Marek, 1999. "Applications of statistical mechanics in number theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 274(1), pages 149-157.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Alexey V. Yakovlev & Vladimir V. Alekseev & Maria V. Volchikhina & Sergey V. Petrenko, 2022. "A Combinatorial Model for Determining Information Loss in Organizational and Technical Systems," Mathematics, MDPI, vol. 10(19), pages 1-12, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Cattani, Carlo & Ciancio, Armando, 2016. "On the fractal distribution of primes and prime-indexed primes by the binary image analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 222-229.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:11:p:1224-:d:563588. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.