Asymptotic prime partitions of integers

Johann Bartel, R. K. Bhaduri, Matthias Brack, and M. V. N. Murthy
Phys. Rev. E 95, 052108 – Published 5 May 2017

Abstract

In this paper, we discuss P(n), the number of ways a given integer n may be written as a sum of primes. In particular, an asymptotic form Pas(n) valid for n is obtained analytically using standard techniques of quantum statistical mechanics. First, the bosonic partition function of primes, or the generating function of unrestricted prime partitions in number theory, is constructed. Next, the density of states is obtained using the saddle-point method for Laplace inversion of the partition function in the limit of large n. This gives directly the asymptotic number of prime partitions Pas(n). The leading term in the asymptotic expression grows exponentially as n/ln(n) and agrees with previous estimates. We calculate the next-to-leading-order term in the exponent, proportional to ln[ln(n)]/ln(n), and we show that an earlier result in the literature for its coefficient is incorrect. Furthermore, we also calculate the next higher-order correction, proportional to 1/ln(n) and given in Eq. (43), which so far has not been available in the literature. Finally, we compare our analytical results with the exact numerical values of P(n) up to n8×106. For the highest values, the remaining error between the exact P(n) and our Pas(n) is only about half of that obtained with the leading-order approximation. But we also show that, unlike for other types of partitions, the asymptotic limit for the prime partitions is still quite far from being reached even for n107.

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  • Received 11 January 2017

DOI:https://doi.org/10.1103/PhysRevE.95.052108

©2017 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

Johann Bartel1,*, R. K. Bhaduri2,†, Matthias Brack3,‡, and M. V. N. Murthy4,§

  • 1Institut Pluridisciplinaire Hubert Curien, Physique Théorique, Université de Strasbourg, F-67037 Strasbourg, France
  • 2Department of Physics and Astronomy, McMaster University, Hamilton, Canada L8S4M1
  • 3Institute of Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany
  • 4The Institute of Mathematical Sciences, Chennai 600 113, India

  • *johann.bartel@iphc.cnrs.fr
  • bhaduri@physics.mcmaster.ca
  • matthias.brack@ur.de
  • §murthy@imsc.res.in

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Vol. 95, Iss. 5 — May 2017

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