24 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…

Complex zero-free-domain conjecture. There exists a complex domain $D_r$ containing the real interval $(2r,\infty)$ such that

Let $G$ be a finite graph of maximum degree $\Delta$, let $Z(w)$ denote its hard-core lattice-gas partition function with equal fugacity $w$, and let $D_\Delta$ be a complex domain…

Let $F$ be a number field, let $obreakboldsymbol{mathbb A}F$ be its ring of adeles, and let $mathfrak{F}{n}$ denote the cuspidal automorphic representations $pi$ of…

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…

Let $\mathfrak{c}$ be the supremum of the constants admissible in Landau's zero-free region, namely … The examples of Diamond, Montgomery, and Vorhauer suggest that the known sharp…

The quasi-Riemann hypothesis. If this claim holds, then in the theorem's setting the auxiliary condition (b) on the coefficients is unnecessary. This is a conditional simplificatio…

Deterministic approximate counting conjecture. Such algorithms should exist provided

Let $\Delta$ be a maximum-degree bound, and let $\mathcal{U}_{\Delta,2}$ be the maximal connected zero-free region containing $0$ for graphs of maximum degree at most $\Delta$. Per…

Let $\mathcal{U}_{\Delta,2}\subseteq\mathbb{C}$ be the maximal open zero-free set for graphs of maximum degree at most $\Delta$, and let…

Let $\mathcal{H}$ be a hypergraph of maximum degree at most $\Delta$, and let $Z(\mathcal{H};\bm{\lambda})$ denote its independence polynomial with complex vertex activities. Defin…

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 $f$ be an entire function of genus one with $f(0)\ne 0$, and let $\kappa$ denote its aperture. If $f$ belongs to the $N$-th order Laguerre–Pólya class, the established bound ha…

Let $G$ be a graph, and let $R_1$ be the zero-free region in the $(q,v)$ plane. A graph is 2-connected if it is connected and has no cut vertex. Jackson–Sokal's hierarchy conjectur…

Zero-free interval conjecture. There exists $\varepsilon>0$ such that, if $G$ is a non-separable graph with a spanning tree of maximum degree $3$, then

Let $L(s,\chi)$ be the Dirichlet $L$-function associated with a real primitive Dirichlet character $\chi$ modulo $q$: … A Siegel zero is a zero of such an $L$-function in the relev…

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

Let $q\geq 1$, let $\chi$ be a character modulo $q$, and write $s=\sigma+it$. Large zero-free region conjecture. There exists a positive constant $c_0'$ sufficiently large such tha…

Large zero-free region conjecture. There exists a constant $c_0>0$ sufficiently large such that, for every $q\geq 1$ and every character $\chi$ modulo $q$, the Dirichlet $L$-functi…