The Distribution of G.C.D.s of Shifted Primes and Lucas Sequences

TL;DR

This paper studies the distribution of the gcd of shifted primes and Lucas sequence terms, providing asymptotic formulas, bounds, and connections to prime gap results, advancing understanding of gcd behavior in number theory.

Contribution

It establishes asymptotic formulas for sums involving gcds of shifted primes and Lucas sequences, and links these to prime gap results and conjectures.

Findings

Asymptotic formulas for sums of powers of gcds over primes.

Upper bounds on the count of primes with large gcds.

Existence of infinitely many prime runs with large gcds.

Abstract

Let $(u_n)_{n \ge 0}$ be a nondegenerate Lucas sequence and $g_u(n)$ be the arithmetic function defined by $\gcd(n, u_n).$ Recent studies have investigated the distributional characteristics of $g_u$. Numerous results have been proven based on the two extreme values $1$ and $n$ of $g_{u}(n)$. Sanna investigated the average behaviour of $g_{u}$ and found asymptotic formulas for the moments of $\log g_{u}$. In a related direction, Jha and Sanna investigated properties of $g_{u}$ at shifted primes. In light of these results, we prove that for each positive integer $\lambda,$ we have $$\sum_{\substack{p\le x\\p\text{ prime}}} (\log g_{u}(p-1))^{\lambda} \sim P_{u,\lambda}\pi(x),$$ where $P_{u, \lambda}$ is a constant depending on $u$ and $\lambda$ which is expressible as an infinite series. Additionally, we provide estimates for $P_{u,\lambda}$ and $M_{u,\lambda},$ where $M_{u, \lambda}$ is the constant for an analogous sum obtained by Sanna [J. Number Theory 191 (2018), 305-315]. As an application of our results, we prove upper bounds on the count $\#\{p\le x : g_{u}(p-1)>y\}$ and also establish the existence of infinitely many runs of $m$ consecutive primes $p$ in bounded intervals such that $g_{u}(p-1)>y$ based on a breakthrough of Zhang, Maynard, Tao, et al. on small gaps between primes. Exploring further in this direction, it turns out that for Lucas sequences with nonunit discriminant, we have $\max\{g_{u}(n) : n \le x\} \gg x$. As an analogue, we obtain that that $\max\{g_u(p-1) : p \le x\} \gg x^{0.4736}$ unconditionally, while $\max\{g_u(p-1): p \le x\} \gg x^{1 - o(1)}$ under the hypothesis of Montgomery's or Chowla's conjecture.