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 $$\sum _{\begin{array}{c}p\le x\\ p\text{ prime}\end{array}}(\log g_u(p-1){)}^{\lambda }\sim P_{u,\lambda }\pi (x),$$ 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.

17 pages; comments welcome