8 problems

GinOE number variance scaling limits. As $N\to\infty$, the following limits are expected. For fixed $a\in(0,1)$,

Let $S(t)$ be the argument function of the Riemann zeta function, and let $\gamma_0$ be Euler's constant. For $\delta>0$, consider the mean square of the increment…

Let $\beta$ be the interaction parameter, and let $c_\beta$ be a beta-dependent constant. The number variance is considered for interacting fermions in a harmonic potential, includ…

Smooth Gaussian-prime variance conjecture. The variance satisfies

Gaussian-prime angle variance conjecture. For $1\ll K\ll N^{1-o(1)}$,

Let $mathcal E(n)$ be the projected lattice points on $S^2$, let $N_n$ be their number, and let $sigma$ denote normalized spherical area. For a spherical cap or annulus…

The Wishart $beta$-ensemble has number variance in the edge regime described by the edge scaling variable $s=(L-4)(N/4)^{2/3}$. The limits $s\to\pm\infty$ are expected to match the…

Let $E_\beta^{(\cdot)}(n;J)$ denote the probability of exactly $n$ particles in $J$, let $n_J$ be the corresponding particle number, and let $\langle n_J\rangle$ and…