12 problems

Hardy Z-function joint moment conjecture. As $T\to\infty$,

Let $\mathbf{A}\in\mathbb{U}(N)$, and let $Z_{\mathbf{A}}(\theta)$ be the real-valued Hardy-type characteristic-polynomial function … For arbitrary positive real parameters $s$ and…

Hybrid product conjecture. Let $k\geq0$ be any real number. Suppose that $X$ and $q\to\infty$ with $X\ll(\log q)^{2-\epsilon}$. Then

Random-matrix conjecture for $Z_X$. Let $k>-\frac{1}{2}$ be any real number. Suppose that $X,q\to\infty$ with $X\ll(\log q)^{2-\epsilon}$ and $w(q)\ll X^{1+\epsilon}$, where $w(q)$…

The Dirichlet moment conjecture. For $k$ fixed with $\operatorname{Re}k\geq0$,

For $m\in\mathbb Z_{\ge1}$, let $F_N(m,h)$ denote the unitary-matrix joint moment introduced in the paper. ABGS divergence conjecture. … This conjecture concerns the rationality qu…

Let $r$ and $\tilde{r}$ be nonnegative integers, let $q>2$ be a prime power, let $s_1,\dots,s_{r+\tilde{r}}$ be complex numbers with real part $\frac{1}{2}$, and let $A$ be a real…

Generalized zeta-derivative moment conjecture. The asymptotic relations stated in the source should hold, in particular

Twisted cubic moment conjecture. The full set of main terms for

Let $k\geqslant1$ be an integer, and let $Q,T\geqslant2$ be real numbers. Let $\mathcal{O}_2(Q)$ denote the family of real Dirichlet characters counted in the paper, and let…

Let … where the sum is over real primitive Dirichlet characters and $D$ tends to infinity. Quadratic Dirichlet moment conjecture. The $k$th moment is conjectured to satisfy … This…

Microscopic logarithmic-derivative moment conjecture. The following two asymptotic identities should hold: