除数问题与$n^2α$小数部分的对相关
Divisor problems and the pair correlation for the fractional parts of $n^2α$
摘要：
Z. Rudnick和P.Sarnak证明了$n^2 α$小数部分的对相关对几乎所有的$α$都是泊松的。然而，他们无法找到一个特定的$α$，它持有。我们发现，这个问题是有关的问题，确定$(a,b,r) \in \N^3$的数量，使$a \le M$，$b \le N$，$r \le K$和$p ab \equiv r (q)$的$p$和$q$互质。通过对$K$、$M$、$N$和$q$的相对大小的适当假设，人们应该期望存在$KMN/q$这样的三重渐近，我们将证明这在平均上成立。
英文摘要：
Z. Rudnick and P. Sarnak have proved that the pair correlation for the fractional parts of $n^2 α$ is Poissonian for almost all $α$. However, they were not able to find a specific $α$ for which it holds. We show that the problem is related to the problem of determining the number of $(a,b,r) \in \N^3$ such that $a \le M$, $b \le N$, $r \le K$ and $p ab \equiv r (q)$ for $p$ and $q$ coprime. With suitable assumptions on the relative size of $K$, $M$, $N$ and $q$ one should expect there to be $KMN/q$ such triples asymptotically and we will show that this holds on average.
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