A generalized Hawkins sieve and prime \(k\)-tuplets (Q2477979)

Let \(\mathfrak{p} : \mathbb{N}_{\geq 2} \to [0, 1]\). In Hawkins' model, a sieving number \(n\) sieves subsequent numbers with probability \(1/n\). In this paper, the author generalizes the model by allowing the sieving numbers \(n\) to sieve subsequent numbers with fixed probability \(\mathfrak{p} (n)\) and calling the resulting sequence a set of Hawkins \(\mathfrak{p}\)-primes. Given certain conditions on the decay of \(\mathfrak{p} (n)\), the author establishes an asymptotic formula for the density of Hawkins \(\mathfrak{p}\)-primes. \textbf{Theorem 1.} Let \(S_{n}\) denote the event that a natural number \(n\) is a sieving number. Suppose that \(\sum \mathfrak{p}^{2} (k)\) converges while \(\sum \mathfrak{p} (k)\) diverges. Then \[ P(S_{n}) \sim \left(\sum_{k \leq n} \mathfrak{p} (k)\right)^{-1}. \] With additional conditions on \(\mathfrak{p} (n)\), the author obtains asymptotic formulas for both \(X_{n}\), the \(n\)-th term of a sequence of Hawkins \(\mathfrak{p}\)-primes, and \(\prod_{k < n} (1 - \mathfrak{p} (X_{k}))^{-1}\), thereby obtaining probabilistic generalizations of the Prime Number Theorem and Mertens' Theorem. Let \(\chi\) denote the set of all strictly increasing sequences of integers larger than 1. For \(n \geq 2\) and \(\alpha \in \chi\), set \[ y_{n} (\alpha) = \prod_{a \in \alpha \atop a < n} (1 - \mathfrak{p} (a)) \] for \(n > 2\), and sets \(y_{2} (\alpha) = 1\). Condition 1. Extend \(\mathfrak{p}\) to a function \(\mathfrak{p} : [2, \infty) \to [0, 1]\) such that (i) \(\mathfrak{p}\) is positive, continuous, and decreasing to zero, with \(\mathfrak{p} (2) < 1\), (ii) \(\sum \mathfrak{p} (n)\) diverges, and (iii) \(\sum \mathfrak{p}^{2} (n)\) converges. Condition 2. Extend \(\mathfrak{p}\) to a continuous function \(\mathfrak{p} : [2, \infty) \to [0, 1]\) such that (i) \(\displaystyle \lim_{a \to 1} \left(\lim_{n \to \infty} \frac{I (a^{n + 1})}{I (a^{n})}\right) = 1\), and (ii) \(\sum (n I (n))^{-1}\) converges, where \(I (n) = \int_{2}^{n} \mathfrak{p} (t) \,dt\). The analog of Mertens' Theorem can be summarized as follows. Corollary. Suppose the sieving probability function \(\mathfrak{p}\) satisfies Conditions 1 and 2. Then \(y_{n} \sim I^{-1} (n)\) a.s. Using this generalization of Mertens' Theorem together with limit theory for orthogonal random variables, the author proves a version of the Prime Number Theorem for Hawkins \(\mathfrak{p}\)-primes. Let \(n \in \mathbb{Z}^{+}\), denote by \(X_{n}\) the random variable on \(\chi\) giving the \(n\)-th generalized prime, and define \(Y_{n} \prod_{k \leq n} (1 - \mathfrak{p} (X_{n}))^{-1}\). A final condition on \(\mathfrak{p}\) is needed. Condition 3. Extend the sieving probability function \(\mathfrak{p}\) to a continuous function \(\mathfrak{p} : [2, \infty) \to [0, 1]\) such that \(n \mathfrak{p} (n) / I (n) \to 0\) as \(n \to \infty\). \textbf{Theorem 2.} Suppose \(\mathfrak{p}\) satisfies Conditions 1, 2, and 3. Then \[ \frac{1}{n} X_{n} \sim Y_{n} \sim I (X_{n}). \] In Theorem 2, the asymptotic equivalence \(X_{n} / n \sim I (X_{n})\) is an analog of the Prime Number Theorem, and \(Y_{n} \sim I (X_{n})\) is another analog of Mertens' Theorem. As an application, the author shows that the sieve produces a new probabilistic model for prime \((k + 1)\)-tuples. \textbf{Theorem 3.} In the case \(\mathfrak{p} (n) = n^{-1} \log^{k} n\), we have \[ X_{n} \sim \frac{n}{k + 1} \log^{k + 1} n \;\text{a.s.} \quad \text{and} \quad Y_{n} \sim \frac{1}{k + 1} \log^{k + 1} n \;\text{a.s.} \]