On the distribution of Hawkins' random ``primes'' (Q1026986)

In this article the author investigates applications of the Hawkins random sieve, a compelling stochastic analog of the sieve of Eratosthenes first introduced about 50 years ago by \textit{D.\ Hawkins} [``The random sieve'', Math. Mag. 31, 1--3 (1957; Zbl 0086.03502)]. Sequences of values produced by the Hawkins sieve are known as Hawkins primes. Using the limit theory of random variables, results have been established for Hawkins primes that parallel many of the famous existing results and conjectures about the distribution of prime numbers in the strongest probabilistic sense. These include the Prime Number Theorem, Mertens' Theorem, the twin primes conjecture, and the Riemann Hypothesis. Notable examples of such results include [\textit{C.\ Heyde}, Proc. Am. Math. Soc. 56, 277--280 (1976; Zbl 0336.60030)], [\textit{W.\ Neudecker}, Math. Proc. Camb. Philos. Soc. 77, 365--367 (1975; Zbl 0312.10034)], and [\textit{M.\ Wunderlich}, Acta Arith. 26, 59--81 (1974; Zbl 0257.10033)]. In the present paper the author proves that versions of the twin primes conjecture and Dirichlet-de la Vallée Poussin's theorem hold almost surely for Hawkins primes. In addition, and perhaps most importantly, the author is able to include error terms for each of the theorems. In establishing his results, the author's point of view is to regard the Hawkins sieve as a Markovian process.