7 problems

Let $Z(t)=\zeta\left(\frac{1}{2}+it\right)e^{-i\theta(t)}$ be the Hardy $Z$-function, and let $g_n$ be the Gram points defined by $\theta(g_n)=\pi n$. A Gram point is bad when it v…

Let $g_n$ be an isolated Gram point, and let $\gamma^{\mathrm{descend}}$ denote the descending curve in the two-dimensional deformation associated with the shifting and descending…

Let $g_n$ be an isolated bad Gram point, meaning the middle point of a Gram block whose adjacent boundary Gram points $g_{n-1}$ and $g_{n+1}$ are good. In the two-dimensional param…

Let $g_n$ be a Gram point, and call it bad when it does not satisfy the usual Gram sign condition; call its consecutive neighbours $g_{n-1}$ and $g_{n+1}$ good when they do satisfy…

Let $E_{SU(N)}(k,\mathcal J)$ denote the probability that a random special unitary matrix has exactly $k$ eigenvalues in the corresponding $SU(N)$ Gram interval $\mathcal J$, and l…

Let $U(N)$ be the group of $N\times N$ unitary matrices, and let $J\subset[-\pi,\pi)$ be an interval of length $2\pi/N$. Define $E_{U(N)}(k,J)$ to be the probability that a random…

Let $t_n$ denote the ordinates of the Gram points, let $gamma_n$ denote the ordinates of the nontrivial zeros of the Riemann zeta function, and for $n\geq 1$ define $\Delta_n=m-n$…