17 problems

Let $T$ be a degenerate exactly-solvable operator, and let $\{q_n\}$ be a family of appropriately scaled eigenpolynomials. Let $\Xi_T$ be the support of the limiting root measure o…

Jacobi polynomials with arbitrary real parameters $\alpha$ and $\beta$ can exhibit hermitian, non-hermitian, or multiple orthogonality, and their zeros may be complex. Let…

Unbounded-support conjecture. In this asymptotically diagonal case, the support of any limit measure $\mu$ is unbounded. The conjecture concerns the degenerate case in which the pa…

Zero-accumulation conjecture. The zeros of the Jacobi–Piñeiro polynomials $P^{(\bm \alpha;\beta)}_{\bm{n},i}$ accumulate on $\overline{\Delta}_r\cup\{\infty\}$ as…

Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ be the open unit disk. For each $n$, consider the unitary Hermite polynomial and its empirical distribution of zeroes, assigning mass…

Angelesco-type vector equilibrium conjecture. In this case, the asymptotic zero distribution can be described in terms of an Angelesco-type vector equilibrium problem.

Let $f_0$ and $f_1$ be the functions associated with disjoint admissible compact sets $F_0\in\mathcal F_0$ and $F_1\in\mathcal F_1$, and let $\Gamma=(\Gamma_0,\Gamma_1)$ be a vecto…

Let $F\in\mathcal F_f$ be an admissible compact set for $f$, and let $\{\Omega_n\}$ be an interpolation table disjoint from $F$. Assume that the limiting density $\omega$ in the in…

Let $\mu$ be a finite Borel measure supported on a compact subset $K$ of the real line and assume that $\mu$ is regular in the sense of Stahl and Totik on $K$. Let…

Minkowski's zero-distribution discrepancy conjecture. As $j$ tends to infinity,

Minkowski's orthogonal-polynomial zero convergence conjecture. As $j$ tends to infinity, the zeros $\zeta_l^j$ converge uniformly to the zeros $\theta_l^j$; moreover, there exist p…

Zero-spacing conjecture. For each $\gamma=(\gamma_k)_{k=1}^\infty$ with $\inf_k\gamma_k>0$, $(A_{2^k})_{k=1}^\infty$ is unbounded. If $s=2^k$ for some $k\in\mathbb{N}$, there is a…

Azaïs et al.'s local universality conjecture. If the coefficients are independent identically distributed with zero mean and finite variance, then the number of real zeros of $X_n$…

Consider the nontrivial zeros of Dirichlet $L$-functions corresponding to primitive characters, and form the multiset of their nonnegative imaginary parts. Linear independence conj…

Let $\Xi_k(z)=\sum_{j=1}^{k}\Phi_j(z)$ and let $\Phi_{k+1}(z)$ be the next summand in the truncated Riemann Xi-function series. For $k\geq1$ and $0\leq t\leq1$, consider…

Let $a>0$ be a fixed real number, and let $Q$ denote the first quadrant of the complex plane. A function has monotonic zeros in $Q$ when its zeros there, listed by increasing real…

Let $Q$ denote the first quadrant of the complex plane, and let $\Xi_{\Delta,N}(z)$ be the truncation after $N$ terms of the hyperbolic-gamma-function series defining the Ramanujan…