The distribution of intermediate prime factors

2023/05/01 by Nathan McNew, Paul Pollack, McNew, Nathan +3 · 1 citation
Mathematics · #11A51 #11N25 #11N37 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2305.01117

openalex publication_date 2023/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P^{(\frac 12)}(n) denote the middle prime factor of n (taking into account multiplicity). More generally, one can consider, for any α∈ (0,1), the α-positioned prime factor of n, P^(α)(n). It has previously been shown that log log P^(α)(n) has normal order αlog log x, and its values follow a Gaussian distribution around this value. We extend this work by obtaining an asymptotic formula for the count of n≤ x for which P^(α)(n)=p, for primes p in a wide range up to x. We give several applications of these results, including an exploration of the geometric mean of the middle prime factors, for which we find that \frac 1x ∑_1

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