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

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.