论文标题
$ x_1^2+x_2^2-x_3^2 = n^2 $的解决方案with small $ x_3 $
Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$
论文作者
论文摘要
Friedlander和Iwaniec调查的集成解决方案$（x_1，x_2，x_3）$他们获得了$ x_3 \ asymp m $的解决方案的渐近公式，其中$ m $远小于$ \ sqrt {d} $。确切地说，它们的状况为$ M \ ge d^{1/2-1/1332} $。他们的分析使他们达到了某些Weyl总和的平均值。 $ d $的条件无平方，对他们的工作至关重要。当$ d = n^2 $是奇数$ n $的平方时，我们调查了“相反”情况。这种情况本质上是不同的，导致了Kloosterman总和。我们获得了$ x_3 \ asymp m $的解决方案的渐近公式，其中$ m \ ge d^{1/2-1/16+\ varepsilon} $用于任何固定的$ \ varepsilon> 0 $。
Friedlander and Iwaniec investigated integral solutions $(x_1,x_2,x_3)$ of the equation $x_1^2+x_2^2-x_3^2=D$, where $D$ is square-free and satisfies the congruence condition $D\equiv 5\bmod{8}$. They obtained an asymptotic formula for solutions with $x_3\asymp M$, where $M$ is much smaller than $\sqrt{D}$. To be precise, their condition is $M\ge D^{1/2-1/1332}$. Their analysis led them to averages of certain Weyl sums. The condition of $D$ being square-free is essential in their work. We investigate the "opposite" case when $D=n^2$ is a square of an odd integer $n$. This case is different in nature and leads to sums of Kloosterman sums. We obtain an asymptotic formula for solutions with $x_3\asymp M$, where $M\ge D^{1/2-1/16+\varepsilon}$ for any fixed $\varepsilon>0$.
