9 problems

Let $n$ be the dimension parameter, let $\rho=(n-1)/2$, and define … where $F(a,b;c;z)$ is the hypergeometric function and $k$ is complex. The zero-free region conjecture. For ever…

Let $Z_{\bm{G}}(\beta)$ denote the partition function of the Sherrington–Kirkpatrick model, let $\varepsilon>0$, and let $\beta_{\mathsf{2nd}}$ be the second-moment threshold. A co…

Galvin–McKinley–Perkins–Sarantis–Tetali conjecture. For each $k\geq 2$, there exists a constant $C_k>0$ such that, if $G$ is a $k$-uniform linear hypergraph of maximum degree…

Zero-free locus conjecture. The zero-free locus of hypergraphs of maximum degree $\Delta$ is identical to the zero-free locus of graphs of maximum degree $\Delta$:

Let $s=\sigma+it$ be the complex variable for a Dirichlet $L$-function. Bentz's conjecture. The domain … is zero free, and there is no zero at $s=\frac12$ for a Dirichlet $L$-funct…

Let $k$ be a modulus, let $\chi$ range over Dirichlet characters modulo $k$, and write $s=\sigma+it$. Haselgrove condition. There is a function $Z(k)$ with $0<Z(k)\leq 1$ such that…

Zero-density conjecture. As $G$ ranges over all graphs, the zeros of $Z_G(q,v)$ are dense in regions (a)--(e). More precisely, for each fixed $q$ the zeros are dense in $v$, or for…

Sign-variation conjecture. Fix real $q$ and $v$ in one of these regions. Then, for all sufficiently large $n$ (depending on $q$ and $v$) and all sufficiently large $m$ (depending o…

Zero-free interval conjecture. Suppose $0<q<1$. Then there exists a strictly increasing sequence of self-dual intervals ${\mathcal{V}}_m(q)$, $m\geq 2$, such that