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Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law from Ecology

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  • Joel E. Cohen

Abstract

Taylor's law, which originated in ecology, states that, in sets of measurements of population density, the sample variance is approximately proportional to a power of the sample mean. Taylor's law has been verified for many species ranging from bacterial to human. Here, we show that the variance V(x) and the mean M(x) of the primes not exceeding a real number x obey Taylor's law asymptotically for large x. Specifically, V(x) ∼ (1/3)(M(x))2 as x → ∞. This apparently new fact about primes shows that Taylor's law may arise in the absence of biological processes, and that patterns discovered in biological data can suggest novel questions in number theory. If the Hardy-Littlewood twin primes conjecture is true, then the identical Taylor's law holds also for twin primes. Taylor's law holds in both instances because the primes (and the twin primes, given the conjecture) not exceeding x are asymptotically uniformly distributed on the integers in [2, x]. Hence, asymptotically M(x) ∼ x/2, V(x) ∼ x2/12. Higher-order moments of the primes (twin primes) not exceeding x satisfy a generalized Taylor's law. The 11,078,937 primes and 813,371 twin primes not exceeding 2 × 108 illustrate these results.

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  • Joel E. Cohen, 2016. "Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law from Ecology," The American Statistician, Taylor & Francis Journals, vol. 70(4), pages 399-404, October.
    • Handle: RePEc:taf:amstat:v:70:y:2016:i:4:p:399-404
    DOI: 10.1080/00031305.2016.1173591
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    Cited by:

    1. Xu, Meng & Jiang, Mengke & Wang, Hua-Feng, 2021. "Integrating metabolic scaling variation into the maximum entropy theory of ecology explains Taylor's law for individual metabolic rate in tropical forests," Ecological Modelling, Elsevier, vol. 455(C).
    2. Meng Xu & Joel E Cohen, 2019. "Analyzing and interpreting spatial and temporal variability of the United States county population distributions using Taylor's law," PLOS ONE, Public Library of Science, vol. 14(12), pages 1-25, December.
    3. Joel E. Cohen & Christina Bohk-Ewald & Roland Rau, 2018. "Gompertz, Makeham, and Siler models explain Taylor's law in human mortality data," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 38(29), pages 773-842.

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