论文标题
关于随机单位矩阵的特征多项式的对数导数
On the logarithmic derivative of characteristic polynomials for random unitary matrices
论文作者
论文摘要
让$ u \ in U（n）$为$ n $的随机统一矩阵，相对于$ u（n）$的HAAR度量分布。令$ p（z）= p_u（z）$为$ u $的特征多项式。我们证明，对于$ z $，可以使用$ p $非常接近$ z $的零来近似于$ \ frac {p'} {p} {p}（z）$，并具有典型的可控错误项。这是Selberg的Riemann Zeta功能的类似物。我们还证明了$ \ frac {p'} {p} {p}（z）$远离单位圆的中央限制定理，这是Zeta lester结果的类似物。
Let $U\in U(N)$ be a random unitary matrix of size $N$, distributed with respect to the Haar measure on $U(N)$. Let $P(z)=P_U(z)$ be the characteristic polynomial of $U$. We prove that for $z$ close to the unit circle, $ \frac{P'}{P}(z) $ can be approximated using zeros of $P$ very close to $z$, with a typically controllable error term. This is an analogue of a result of Selberg for the Riemann zeta-function. We also prove a mesoscopic central limit theorem for $ \frac{P'}{P}(z) $ away from the unit circle, and this is an analogue of a result of Lester for zeta.
