论文标题
转移的素数和卢卡斯序列的G.C.D.S分布
The Distribution of G.C.D.s of Shifted Primes and Lucas Sequences
论文作者
论文摘要
令$（u_n）_ {n \ ge 0} $为非等级卢卡斯序列，$ g_u（n）$是$ \ gcd（n，u_n）定义的算术函数。$最近的研究调查了$ g_u $的分销特性。根据两个极端值$ 1 $和$ n $的$ g_ {u}（n）$证明了许多结果。 Sanna研究了$ g_ {u} $的平均行为，并在$ \ log g_ {u} $的时刻找到了渐近公式。在相关的方向上，Jha和Sanna调查了$ g_ {u} $的属性。鉴于这些结果，我们证明，对于每个正整数$λ，$，我们都有$$ \ sum _ {\ or x {p \ le x \\ p \ p \ p \ p \ p \ text {prime}}}（\ log g_ {u g_ {u}（u g_ {u}（p-1）（p-1）（p-1） $ u $和$λ$，可作为无限系列。此外，我们还提供$ p_ {u，λ} $和$ m_ {u，λ}的估计值，其中$ m_ {u，λ} $是Sanna [J.数字理论191（2018），305-315]。 As an application of our results, we prove upper bounds on the count $\#\{p\le x : g_{u}(p-1)>y\}$ and also establish the existence of infinitely many runs of $m$ consecutive primes $p$ in bounded intervals such that $g_{u}(p-1)>y$ based on a breakthrough of Zhang, Maynard, Tao, et al.在素数之间的小差距上。在这个方向上进一步探索，事实证明，对于具有nonit Indictant的Lucas序列，我们有$ \ max \ {g_ {u}（n）：n \ le x \} \ gg x $。作为一个类似物，我们得到了$ \ max \ {g_u（p-1）：p \ le x \} \ gg x^{0.4736} $无条件，而$ \ max \ {g_u（p-1）：p \ le x \}猜想。
Let $(u_n)_{n \ge 0}$ be a nondegenerate Lucas sequence and $g_u(n)$ be the arithmetic function defined by $\gcd(n, u_n).$ Recent studies have investigated the distributional characteristics of $g_u$. Numerous results have been proven based on the two extreme values $1$ and $n$ of $g_{u}(n)$. Sanna investigated the average behaviour of $g_{u}$ and found asymptotic formulas for the moments of $\log g_{u}$. In a related direction, Jha and Sanna investigated properties of $g_{u}$ at shifted primes. In light of these results, we prove that for each positive integer $λ,$ we have $$\sum_{\substack{p\le x\\p\text{ prime}}} (\log g_{u}(p-1))^λ \sim P_{u,λ}π(x),$$ where $P_{u, λ}$ is a constant depending on $u$ and $λ$ which is expressible as an infinite series. Additionally, we provide estimates for $P_{u,λ}$ and $M_{u,λ},$ where $M_{u, λ}$ 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.
