401 problems

Let $1/2<\sigma<1$ be fixed. Montgomery's lower bound concerns the maximum of $\log|\zeta(\sigma+it)|$ for $t\in[T,2T]$ as $T\to\infty$: … where $c(\sigma)>0$. Montgomery's conject…

Strict monotonicity conjecture. The function $\boldsymbol D$ is strictly decreasing on $(0,1)$; in particular, $\boldsymbol\rho_0$ is its unique real zero.

Let $p$ be a prime and let $f\in\mathbb{Z}[x_1,\dots,x_d]$. The poles of the Igusa zeta function encode the asymptotic behavior of the numbers of solutions of $f\equiv 0\pmod{p^k}$…

Analytic continuation conjecture. The function $\zeta_{\mathrm{ts}}(s;G,H)$ admits an analytic continuation to the complex plane, with poles reflecting the arithmetic structure of…

Let $g_k$ denote the universal factor in the conjectural leading constant for the $2k$th moment of the Riemann zeta function. Conrey–Gonek conjecture. The eighth moment constant is…

Let $F:(\mathbb{C}^{n+1},\mathbf{0})\to(\mathbb{C},\mathbf{0})$ define a germ of an isolated hypersurface singularity. For $x\in F^{-1}(0)$, let $\mathcal{F}_{F,x}$ be the Milnor f…

Weil's conjecture. The rational function $Z(X_p,T)$ has the form

Let $\mathbb{F}_q$ be a finite field and let $n\geq 1$. Write $\zeta(\mathbb{F}_q,s)$ for the zeta function of $\mathbb{F}_q$, and let $\mathrm{K}_i(\mathbb{F}_q)$ denote its algeb…

Let $I_k(T)$ denote the $2k$th moment of the Riemann zeta function, and write its conjectural leading constant as $C_k=g_k a_k/\Gamma(k^2+1)$, where $a_k$ is the arithmetic factor.…

Berry–Keating conjecture. When expanded to its optimal order, the Riemann–Siegel formula should approximate $Z(t)$ with an error of order

Let $\zeta(s)$ be the Riemann zeta function. Its nontrivial zeros are the zeros in the critical strip $0<\operatorname{Re}(s)<1$. Simplicity conjecture. All nontrivial zeros of…

Let $\mathbb{K}$ be a number field, let $\zeta_{\mathbb{K}}(s)$ be its Dedekind zeta function, and call a zero nontrivial if it lies in the critical strip $0<\operatorname{Re}s<1$.…

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…

Let $\mathcal{L}$ be a nilpotent Lie lattice considered in CSV/19, let $K$ be a number field, and for each decomposition type $(\mathbf{e},\mathbf{f})$ let…

Irrationality conjecture for odd zeta values. For every $n\in\mathbb{N}$, $\zeta(2n+1)$ is irrational.

Let $a\in\mathbb{C}$ be fixed, and consider the roots of $\zeta(s)-a$. Selberg's conjecture. There are only finitely many roots of $\zeta(s)-a$ on the critical line…

Let $f$ be a polynomial, and let the complexification of $f$ have monodromy eigenvalues. Consider the Igusa zeta function of $f$ with respect to a character, and let the order of t…

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…

Let $K$ be a complex cubic field, let $K(\mathfrak{f})$ be the relevant narrow ray class field, let $e_{\mathfrak{f}}$ be the number of roots of unity in $K(\mathfrak{f})$, and fix…

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…

Folklore conjecture. As $T\to\infty$, there is a positive constant $C_k$ such that

Let $\alpha$ be a $\mathbb{Z}$-action generated by an ergodic automorphism of a compact abelian group, and let $\zeta_\alpha$ denote its dynamical zeta function. Natural-boundary c…

Let $B_n$ be the $n$-th Bernoulli number. Sun's conjecture. The sequence … is strictly increasing. The paper proves this, confirming Sun's conjecture; Luca and Stănică independentl…

The Riemann zeta function is denoted by $\zeta$. The zeros of $\zeta$ are the complex numbers at which it vanishes, and a large Hermitian random matrix is a random matrix whose eig…

Let $k$ be a positive integer, let $\zeta^{(k)}$ denote the $k$-th derivative of the Riemann zeta function, and let $a_k$ be the arithmetic factor appearing in the conjectural form…