论文标题
关于卢卡斯序列的外观索引
On the index of appearance of a Lucas sequence
论文作者
论文摘要
令$ \ mathbf {u} =（u_n）_ {n \ geq 0} $为lucas序列，也就是说，满足$ u_0 = 0 $，$ u_1 = 1 $的整数序列$ a_1 $和$ a_2 $是固定的非零整数。对于每个素数$ p $，带有$ p \ nmid 2a_2d _ {\ mathbf {u}} $，其中$ d _ {\ mathbf {u}}：= = a_1^2 + 4a_2 $，$ d + 4a_2 $，$ p _ {\ mathbf {u}} $ s $ semp of speck也就是说，最小的正整数$ k $，使得$ p \ mid u_k $。众所周知，存在$ρ_ {\ mathbf {u}}（p）$存在，并且$ p \ equiv \ big（d _ {\ mathbf {u}} \ mid p \ big）\ big）\ pmod {ρ_ $ \ big（d _ {\ mathbf {u}} \ mid p \ big）$是legendre符号。将$ p $在$ \ MathBf {U}中的外观定义为$ _ {\ MathBf {u}}}（p）：= \ left（p- \ big（d _ {\ MathBf {u}}} \ mid p \ big）对于每个积极的整数$ t $，对于每$ x> 0 $，让$ \ Mathcal {p} _ {\ MathBf {u}}}（t，x）$是质数$ p $的集合，以便$ p \ leq x $，$ p \ leq x $，$ p \ nmid 2a_2 d _2 d _2 d _ { $ | _ {\ Mathbf {u}}（p）= t $。在广义的Riemann假设下，在$ \ Mathbf {U} $上的一些温和假设下，我们证明\ begin {equation {equation*} \＃\ Mathcal {p} _ {\ MathBf {u}}（t，x）= a \，f _ {\ Mathbf {u}}}（t）\，g _ {\ Mathbf { o _ {\ MathBf {u}} \！\ left（\ frac {x} {（\ log x）^2} + \ frac {x \ log（2 \ log x）} {φ（φ（T）（\ log x）^2} \ right） $ a $是artin常数，$ f _ {\ mathbf {u}}（\ cdot）$是乘法函数，而$ g _ {\ mathbf {u}}（\ cdot）$是一个周期函数（这两个函数都可以在$ \ mathbf {u} $中有效地计算。此外，我们提供了一些明确的示例和数值数据。
Let $\mathbf{u} = (u_n)_{n \geq 0}$ be a Lucas sequence, that is, a sequence of integers satisfying $u_0 = 0$, $u_1 = 1$, and $u_n = a_1 u_{n - 1} + a_2 u_{n - 2}$ for every integer $n \geq 2$, where $a_1$ and $a_2$ are fixed nonzero integers. For each prime number $p$ with $p \nmid 2a_2D_{\mathbf{u}}$, where $D_{\mathbf{u}} := a_1^2 + 4a_2$, let $ρ_{\mathbf{u}}(p)$ be the rank of appearance of $p$ in $\mathbf{u}$, that is, the smallest positive integer $k$ such that $p \mid u_k$. It is well known that $ρ_{\mathbf{u}}(p)$ exists and that $p \equiv \big(D_{\mathbf{u}} \mid p \big) \pmod {ρ_{\mathbf{u}}(p)}$, where $\big(D_{\mathbf{u}} \mid p \big)$ is the Legendre symbol. Define the index of appearance of $p$ in $\mathbf{u}$ as $ι_{\mathbf{u}}(p) := \left(p - \big(D_{\mathbf{u}} \mid p \big)\right) / ρ_{\mathbf{u}}(p)$. For each positive integer $t$ and for every $x > 0$, let $\mathcal{P}_{\mathbf{u}}(t, x)$ be the set of prime numbers $p$ such that $p \leq x$, $p \nmid 2a_2 D_{\mathbf{u}}$, and $ι_{\mathbf{u}}(p) = t$. Under the Generalized Riemann Hypothesis, and under some mild assumptions on $\mathbf{u}$, we prove that \begin{equation*} \#\mathcal{P}_{\mathbf{u}}(t, x) = A\, F_{\mathbf{u}}(t) \, G_{\mathbf{u}}(t) \, \frac{x}{\log x} + O_{\mathbf{u}}\!\left(\frac{x}{(\log x)^2} + \frac{x \log (2\log x)}{φ(t) (\log x)^2}\right) , \end{equation*} for all positive integers $t$ and for all $x > t^3$, where $A$ is the Artin constant, $F_{\mathbf{u}}(\cdot)$ is a multiplicative function, and $G_{\mathbf{u}}(\cdot)$ is a periodic function (both these functions are effectively computable in terms of $\mathbf{u}$). Furthermore, we provide some explicit examples and numerical data.
