Dr Sacha Mangerel alexander.mangerel@durham.ac.uk
Assistant Professor
On the bivariate Erdős–Kac theorem and correlations of the Möbius function
Mangerel, ALEXANDER P.
Authors
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 |
You might also like
Three conjectures about character sums
(2023)
Journal Article
Multiplicative functions in short arithmetic progressions
(2023)
Journal Article
Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments
(2023)
Journal Article
Divisor-bounded multiplicative functions in short intervals
(2023)
Journal Article
Short Character Sums and the Pólya–Vinogradov Inequality
(2022)
Journal Article
Downloadable Citations
About Durham Research Online (DRO)
Administrator e-mail: dro.admin@durham.ac.uk
This application uses the following open-source libraries:
SheetJS Community Edition
Apache License Version 2.0 (http://www.apache.org/licenses/)
PDF.js
Apache License Version 2.0 (http://www.apache.org/licenses/)
Font Awesome
SIL OFL 1.1 (http://scripts.sil.org/OFL)
MIT License (http://opensource.org/licenses/mit-license.html)
CC BY 3.0 ( http://creativecommons.org/licenses/by/3.0/)
Powered by Worktribe © 2024
Advanced Search