The shifted prime-divisor function over shifted primes

2024/06/07 by Steve Fan, Fan, Steve · 3 citations
Mathematics · #11N37 #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary: 11N36 #Rings, Modules, and Algebras #Secondary: 11B05

paper · pdf · doi:10.48550/arxiv.2406.05217

openalex publication_date 2024/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let a,b∈ℤ∖\0\. For every n∈ℕ, denote by ω_a^*(n) the number of shifted-prime divisors p-a of n, where p>a is prime. In this paper, we study the moments of ω_a^* over shifted primes p-b. Specifically, we prove an asymptotic formula for the first moment and upper and lower bounds of the correct order of magnitude for the second moment. These results suggest that the average behavior of ω^*_a on shifted primes is similar to its average behavior on natural numbers. We shall also prove upper bounds for the mean values of sub-multiplicative functions in a nice class over the least common multiples of the shifted primes p-a and q-b. Such upper bounds are intimately related to the second moments of ω^*_a over natural numbers and over shifted primes. Finally, we propose a new conjecture on the second moment of ω₁^* over natural numbers and provide a heuristic argument in support of this conjecture.

Cited by

• The maximal order of the shifted-prime divisor function
• The Hardy--Ramanujan inequality for sifted sets and its applications
• Moments of the shifted prime divisor function

Related

• Shifted-prime divisors
• Moments of the shifted prime divisor function
• The maximal order of the shifted-prime divisor function
• On the largest square divisor of shifted primes
• Poisson Approximation of prime divisors of shifted primes