176 problems

Problem. Determine $\kappa$. In particular, is

Let $\overline{\zeta}_n$ be the symmetrized Mordell–Tornheim zeta value of order $n$, and let $p(n)$ denote the partition function. A monomial has total weight $n$ when the sum of…

Let $\rho=\beta+i\gamma$ and $\rho'=\beta'+i\gamma'$ range over the nontrivial zeros of the Riemann zeta function. For fixed $\lambda>0$ and $T\to\infty$, consider pairs with posit…

Let $k=\mathbb F_q$ be a finite field, let $T=\operatorname{Spec} k$, and let $X$ be a smooth proper surface over $T$. Let $P_2(X,t)$ be the characteristic polynomial of Frobenius…

Let $k>0$ be fixed, and let $T$ tend to infinity with $[?]$—the integrals below are taken along the line $[?]$. Gonek's conjecture. Uniformly for $1\leq\delta\leq\log T$, … while u…

Let $f \in \mathbb{C}[x_1,\dots,x_n] \setminus \mathbb{C}$ and assume $f(o)=0$. The local motivic zeta function $Z_{\operatorname{mot},o}(f;s)$ and the local topological zeta funct…

Let $V_{\mathbb{A}_{1,k}}$ denote the volume of the moduli space of $(1,k)$-annuli, viewed as a polynomial in $\pi^2$. Here $k$ is a positive integer. Leading-term conjecture for a…

Broadhurst–Mellit conjecture. One has

For a positive integer $n$, let $q$ be the residue-field cardinality and let $s$ be a complex variable. Let $\mathcal{Z}_{n,n,\mathfrak{o}}^{\mathsf{C}}(s)$ denote the symplectic l…

Strong twisted pair correlation conjecture. Fix $\varepsilon>0$. Uniformly for all prime powers $n\leq T^{1-\varepsilon}$,

Let $k,\gamma\in\mathbb{N}$ and $c>0$ be fixed, and define … Let $a_{k,\gamma}$ be the arithmetic constant and $g_{k,\gamma}$ the geometric constant. Bailey–Keating conjecture. … M…

Let $\mathfrak{f}_{c,d}$ denote the free class-$c$ nilpotent Lie rings on $d$ generators defined over $\mathbb{Z}$. For $c,d\in\mathbb{N}$, let…

Soient $F$ un corps de nombres et $n\geq 2$. Notons $d_n=r_1+r_2$ si $n$ est impair et $d_n=r_2$ si $n$ est pair, et soit…

Let $V_d:=\Gamma(\mathbb{P}^n,\mathcal{O}(d))$ and let $U_d\subset V_d$ be the space of smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$. Projective-space Hadamard convergence…

Let ${\mathcal X}$ be a regular scheme of pure dimension $d$ proper over $\operatorname{Spec}(\mathbb Z)$, and let $n\in\mathbb Z$. Assume that ${\mathcal X}$ satisfies Assumptions…

Functional equation conjecture. The function $\zeta(\mathcal X,s)$ should have a meromorphic continuation to all $s\in\mathbb C$ and satisfy

Let $\mathcal{S}\to\operatorname{Spec}(\mathcal{O}_k)$ be a proper, regular model of a smooth projective curve over a number field $k$. For a finite set of finite Galois extensions…

Let $I$ be a finite category with series Euler characteristic. Write $A_I$ for its adjacency matrix and $\chi_{\Sigma}(I)$ for its series Euler characteristic. The zeta function of…

Egami–Matsumoto's effective independence conjecture. There exists some $\alpha<\frac{\pi}{2}$ such that either $\{\gamma_1,\gamma_2\}=\{\gamma_3,\gamma_4\}$, or

Corrected Ruelle zeta-function conjecture. For any $x_1,\dots,x_k\in\mathbb F_q\setminus\{0,1,t\}$ with $k\ge 1$, the series

Let $X$ be a scheme of finite type over $\mathbb{Z}$, let $d=\dim X$, and assume that higher Chow groups $CH^r(X,i)$ are defined for such schemes, have finite ranks, and satisfy…

The Frobenius matrix $F(\Gamma)$ is the matrix arising from the Frobenius action in the recursive computation of the zeta function, with parameter $\Gamma$. At infinity, a function…

Let $S\subseteq\mathbb{P}^d$ be a possibly singular del Pezzo surface, let $U\subseteq S$ be obtained by deleting exceptional divisors, and let $\rho_S$ be the relevant Picard rank…

Lichtenbaum–Milne conjecture. For every relevant integer $r$, as $t\to q^{-r}$,

Let $t=\varphi(s)$ be a meromorphic function on $\mathbb{C}$, and let $\zeta_{q,r}(s,t,z;\boldsymbol{\omega})$ and $\zeta_r(s,z;\boldsymbol{\omega})$ denote the corresponding $q$-a…