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

ASBTRACT

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.

KEYWORDS:

• Generalized Taylor's law
• Power law
• Prime number theorem
• Twin prime conjecture
• Uniform distribution
• Variance function

Acknowledgment

The author thanks Lek-Heng Lim, Carl Pomerance, the editor Nicole Lazar, and the associate editor for helpful exchanges and suggestions, and Priscilla K. Rogerson for help.

Funding

This work was partially supported by U.S. National Science Foundation grant DMS-1225529.