2023/10/23 by Benjamin Durkan, Durkan, Benjamin
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR) #Statistical Distribution Estimation and Applications
paper · pdf · doi:10.48550/arxiv.2310.14760
openalex publication_date 2023/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note we prove an effective version of the Hardy--Ramanujan Theorem. For every x≥ 2 and every non-negative function F on the non-negative integers, we show (1)/(x)∑2≤ n≤ xF(ω(n)-1)≤ 118 𝔼F(Zloglog x+4.096), where Zλ is Poisson with parameter λ. Thus the shifted empirical distribution of ω(n) is pointwise dominated by a fixed multiple of a Poisson law. We also obtain the sharper squarefree analogue, derive explicit Chernoff and Gaussian-window estimates, obtain moderate-deviation upper bounds and uniform moment estimates, and transfer these consequences to Ω(n) and to the number of prime divisors occurring exactly once.