The shifted prime-divisor function over shifted primes

TL;DR

This paper investigates the statistical behavior of a shifted prime-divisor function over shifted primes, providing asymptotic formulas, bounds, and conjectures that relate its behavior on primes to that on natural numbers.

Contribution

It introduces new asymptotic results and bounds for the moments of the shifted prime-divisor function, and proposes a conjecture linking its behavior on natural numbers and shifted primes.

Findings

Asymptotic formula for the first moment of the function.

Bounds for the second moment of the function.

A new conjecture on the second moment over natural numbers.

Abstract

Let $a,b\in\mathbb{Z}\setminus\{0\}$. For every $n\in\mathbb{N}$, denote by $\omega_a^*(n)$ the number of shifted-prime divisors $p-a$ of $n$, where $p>a$ is prime. In this paper, we study the moments of $\omega_a^*$ 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 $\omega^*_a$ 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 $\omega^*_a$ over natural numbers and over shifted primes. Finally, we propose a new conjecture on the second moment of $\omega_1^*$ over natural numbers and provide a heuristic argument in support of this conjecture.