Asymptotic independence of Ω(n) and Ω(n+1) along logarithmic averages

2024/12/23 by Dimitrios Charamaras, Charamaras, Dimitrios, Florian K. Richter +1
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2412.17583

Abstract

Let Ω(n) denote the number of prime factors of a positive integer n counted with multiplicities. We show that for any bounded functions a,b\colonℕ→ℂ, \frac1logN∑_n=1^N (a(Ω(n))b(Ω(n+1)))/(n) = ((1)/(N)∑_n=1^N a(Ω(n)))((1)/(N)∑_n=1^N b(Ω(n))) + o_N→∞(1). This generalizes a theorem of Tao on the logarithmically averaged two-point correlation Chowla conjecture. Our result is quantitative and the explicit error term that we obtain establishes double-logarithmic savings. As an application, we obtain new results about the distribution of Ω(p+1) as p ranges over ℓ-almost primes for a "typical" value of ℓ.

Related

• The logarithmically averaged Chowla and Elliott conjectures for two-point correlations
• Bilinear obstructions in the cancellation theory of multiplicative functions
• On the arithmetic average of the first n primes
• Improved bounds for the two-point logarithmic Chowla conjecture
• On the Bivariate Erdős-Kac Theorem and Correlations of the Möbius Function