论文标题
$ \ mathbb {f} _q [t] $：i的两个间隔的两个正方形的总和的差异
The Variance of the Sum of Two Squares over Intervals in $\mathbb{F}_q [T]$: I
论文作者
论文摘要
对于$ b \ in \ mathbb {f} _q [t] $ $ $ 2n \ geq 2 $，请考虑写作$ b = e^2 +γf^2 $的数量$ \ mathrm {deg} \ hspace {0.25em} e = n $和$ \ mathrm {deg} \ hspace {0.25em} f = m <n $。我们用$ s_ {γ表示此表示； M}（b）$。我们获得了$ s_ {γ的方差的精确公式； m}（b）$ bo $ \ mathbb {f} _q [t] $中的间隔$。我们使用作者先前用于除数函数的方差和相关性的添加字符和Hankel矩阵的方法。在第2节中，我们简要概述了我们的方法；然后，我们简要讨论结果可能扩展到编写$ b = e^2 + t f^2 $的方式的数量。
For $B \in \mathbb{F}_q [T]$ of degree $2n \geq 2$, consider the number of ways of writing $B=E^2 + γF^2$, where $γ\in \mathbb{F}_q^*$ is fixed, and $E,F \in \mathbb{F}_q [T]$ with $\mathrm{deg} \hspace{0.25em} E = n$ and $\mathrm{deg} \hspace{0.25em} F = m < n$. We denote this by $S_{γ; m} (B)$. We obtain an exact formula for the variance of $S_{γ; m} (B)$ over intervals in $\mathbb{F}_q [T]$. We use the method of additive characters and Hankel matrices that the author previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach; and we briefly discuss the possible extension of our result to the number of ways of writing $B=E^2 + T F^2$.
