On Certain Moments of the Logarithmic Derivative of Characteristic Polynomials of Random Unitary Matrices

The moments for the logarithmic derivative of characteristic polynomials for random unitary matrices are calculated in the ``macroscopic" and ``mesoscopic" ranges at distance from the unit circle. For matrices in the unitary group $U(N)$, the $2K^{\mathrm{th}}$ moments on circles of radius $1-\frac{a}{N}$ for applicable parameters $a$ are found to be asymptotically $\frac{K!}{\left(2-\frac{a}{N}\right)^{2K}}\left(\frac{N}{a}\right)^{2K}$ as $N\rightarrow\infty$. The possibilities and challenges of extending these methods to other ranges of the moment are also discussed.