Given a positive integer n let ω (n) denote the number of distinct prime factors of n, and let a be fixed positive integer. Extending work of Kubilius, we develop a bivariate probabilistic model to study the joint distribution of the deterministic vectors (ω(n), ω(n + a)), with n ≤ x as x → ∞, where n and n + a belong to a subset of ℕ with suitable properties. We thus establish a quantitative version of a bivariate analogue of the Erdős–Kac theorem on proper subsets of ℕ.We give three applications of this result. First, if y = x0(1) is not too small then we prove (in a quantitative way) that the y-truncated Möbius function μy has small binary autocorrelations. This gives a new proof of a result due to Daboussi and Sarkőzy. Second, if μ(n; u) :=e(uω(n)), where u ∈ ℝ then we show that μ(.; u) also has small binary autocorrelations whenever u = o(1) and $u\sqrt {\mathop {\log }\nolimits_2 x} \to \infty$, as x → ∞. These can be viewed as partial results in the direction of a conjecture of Chowla on binary correlations of the Möbius function.Our final application is related to a problem of Erdős and Mirsky on the number of consecutive integers less than x with the same number of divisors. If $y = x^{{1 \over \beta }}$, where β = β(x) satisfies certain mild growth conditions, we prove a lower bound for the number of consecutive integers n ≤ x that have the same number of y-smooth divisors. Our bound matches the order of magnitude of the one conjectured for the original Erdős-Mirsky problem.

中文翻译：

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关于二元 Erdős-Kac 定理和莫比乌斯函数的相关性

给定一个正整数n让ω(n) 表示不同质因数的数量n， 然后让一种为固定正整数。扩展 Kubilius 的工作，我们开发了一个双变量概率模型来研究确定性向量的联合分布（ω(n),ω(n+一种））， 和n≤X作为X→ ∞，其中n和n+一种属于具有合适属性的ℕ子集。因此我们建立了一个定量的Erdős–Kac 定理在 ℕ 的真子集上的二元模拟版本。我们给出了这个结果的三个应用。首先，如果是的=X0(1)不是太小，那么我们（以定量的方式）证明是的-截断莫比乌斯函数μ是的有小的二元自相关。这为 Daboussi 和 Sarkőzy 的结果提供了新的证明。其次，如果μ(n;你) :=e(uω(n））， 在哪里你∈ ℝ 那么我们证明μ(.;你) 也有小的二元自相关你=○(1) 和$u\sqrt {\mathop {\log }\nolimits_2 x} \to \infty$， 作为X→ ∞。这些可以看作是 Chowla 关于莫比乌斯函数二元相关性猜想方向的部分结果。我们的最终应用与 Erdős 和 Mirsky 关于连续整数个数小于X具有相同数量的除数。如果$y = x^{{1 \over \beta }}$, 其中 β = β(X) 满足某些温和的增长条件，我们证明了连续整数个数的下限n≤X具有相同数量的是的-平滑除数。我们的界限与对原始 Erdős-Mirsky 问题的猜想的数量级相匹配。