2016/12/30 by Alexander P. Mangerel, Mangerel, Alexander P.
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1612.09544
openalex publication_date 2016/12/30 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
Let 2 ≤ y ≤ x such that β:= (log x)/(log y) → ∞. Let ωy(n) denote the number of distinct prime factors p of n such that p ≤ y, and let μy(n) := μ2(n)(-1)ωy(n), where μ is the Möbius function. We prove that if β is not too large (in terms of x) then for each fixed a ∈ ℕ, ∑n ≤ x μy(n)μy(n+a) ≪ x((1)/(log2 y) + e-(1)/(21)βlog β). This can be seen as a partial result towards the binary Chowla conjecture. Our main input is a quantitative bivariate analogue of the Erdős-Kac theorem regarding the distribution of the pairs (ω(n),ω(n+a)), where n and n+a both belong to any subset of the positive integers with suitable sieving properties; moreover, we show that the set of squarefree integers is an example of such a set. We end with a further application of this probabilistic result related to a problem of Erdős and Mirsky on the number of integers n ≤ x such that τ(n) = τ(n+1).