论文标题
双变量二次序列的最不常见的倍数
The Least Common Multiple of a Bivariate Quadratic Sequence
论文作者
论文摘要
令$ f \ in \ mathbb {z} [x，y] $是第二个学位的多项式。在本文中，我们发现$ f $ f $ f $ n $的最小常见倍数的渐近行为最高$ n $。更准确地说，我们考虑$ψ_f（n）= \ log \ left（\ text {lcm} _ {0 <f（x，x，y）\ leq n} \ left \ lbrace f（x，x，x，y）\ right \ right \ rbrace \ right）$ n $ tyn $ tyn $ nd $ ndriend to n n $ niftion。事实证明，根据$ f $，有4种不同的渐近行为。对于通用$ f $，我们显示函数$ψ_f（n）$具有$ \ frac {n \ log \ log \ log n} {\ sqrt {\ sqrt {\ log n}} $。我们还表明，根据合适的随机模型，这是预期的数量级。但是，特殊的多项式$ f $可以具有不同的行为，有时会偏离随机模型。我们完整地描述了这些可能行为的数量级，以及每种行为发生时。
Let $F\in\mathbb{Z}[x,y]$ be some polynomial of degree 2. In this paper we find the asymptotic behaviour of the least common multiple of the values of $F$ up to $N$. More precisely, we consider $ψ_F(N) = \log\left(\text{LCM}_{0<F(x,y)\leq N}\left\lbrace F(x,y)\right\rbrace\right)$ as $N$ tends to infinity. It turns out that there are 4 different possible asymptotic behaviours depending on $F$. For a generic $F$, we show that the function $ψ_F(N)$ has order of magnitude $\frac{N\log\log N}{\sqrt{\log N}}$. We also show that this is the expected order of magnitude according to a suitable random model. However, special polynomials $F$ can have different behaviours, which sometimes deviate from the random model. We give a complete description of the order of magnitude of these possible behaviours, and when each one occurs.
