On the bivariate Erdős–Kac theorem and correlations of the Möbius function

Mangerel, ALEXANDER P.

Abstract

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 , 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 , 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.

Citation

Mangerel, A. P. (2019). On the bivariate Erdős–Kac theorem and correlations of the Möbius function. Mathematical Proceedings of the Cambridge Philosophical Society, 169(3), 547-605. https://doi.org/10.1017/s0305004119000288

• Journal Article Type: Article
• Acceptance Date: Jul 14, 2019
• Online Publication Date: Aug 14, 2019
• Publication Date: Aug 14, 2019
• Deposit Date: Oct 20, 2021
• Journal: Mathematical Proceedings of the Cambridge Philosophical Society
• Print ISSN: 0305-0041
• Electronic ISSN: 1469-8064
• Publisher: Cambridge University Press
• Volume: 169
• Issue: 3
• Pages: 547-605
• DOI: https://doi.org/10.1017/s0305004119000288
• Public URL: https://durham-repository.worktribe.com/output/1230984