Random matrix theory and the zeros of ζ′(s)
Random matrix theory and the zeros of ζ′(s)
复制标题
随机矩阵理论和 ζ′(s) 的零点
DOI：
--
复制
发表时间：
2002
期刊：
影响因子：
--
通讯作者：
F. Mezzadri
中科院分区：
文献类型：
--
作者：
F. Mezzadri
研究了由Haar测度给出分布的N × N随机酉矩阵的特征多项式Z（U，z）的导数的根在酉群上的稠密性.基于先前的黎曼zeta函数的随机矩阵理论模型，预计这是对临界线右侧的零点的水平分布的精确描述。我们证明了当N → ∞时，Z '（U，z）的根在区域1 − x/（N − 1）≤| z| < 1趋于极限函数。我们推导了该函数在极限x → ∞和x → 0下的渐近表达式，并与数值实验进行了比较。
We study the density of the roots of the derivative of the characteristic polynomial Z(U, z) of an N × N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function ζ(s), this is expected to be an accurate description for the horizontal distribution of the zeros of ζ'(s) to the right of the critical line. We show that as N → ∞ the fraction of the roots of Z'(U, z) that lie in the region 1 − x/(N − 1) ≤ |z| < 1 tends to a limit function. We derive asymptotic expressions for this function in the limits x → ∞ and x → 0 and compare them with numerical experiments.