6. Other

Problem 6.1.
Given a function whose logarithmic derivative satisfies a third order ODE of Chazy type, what can one say about the zeros of its Fourier transform? For example, are they all real? 
Problem 6.2.
For general $\beta$, can one prove that the polynomials related to the coefficients of \[\mathbb{E}_{G\beta E}\operatorname{Tr} X^k\equiv Nc_{1,k}(\beta)+\cdots+N^{k+1}c_{k+1,k}(\beta),\] have zeros on the unit circle?
Cite this as: AimPL: Painleve equations and their applications, available at http://aimpl.org/painleveapp.