64 problems

Let $\Phi$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $\Xi$-function. Define $s(t)=\Phi(\sqrt{t})$ and $f(t)=s'(t)^2-s(t)s''(t)$ for $t>0$. T…

For every integer $n\ge 1$, every $-1<\alpha<\beta$, and every $0<c\le 1$, let $\pi_n$ be the monic degree-$n$ polynomial orthogonal on $[-c,c]$ with respect to the Jacobi weight…

Let $U$ be a bounded open set in ${\mathbb R}^d$. Write $C(\overline U)$ for the space of continuous real-valued functions on $\overline U$, and let $G_\delta$ denote a countable i…

A unit speed curve is a curve parametrized by arclength, and a $C^1$ curve has a continuous first derivative. Existence conjecture. There is a continuous function … whose restricti…

Nonvanishing normal derivative conjecture. For every integer $t\geq 0$,

Let $\left\{a_n^{(\omega)}\right\}_{n=1}^{\infty}$ range over all possible sequences, and let $p\tilde{\lim}a_n^{(\omega)}$ denote the set of approximated partial limits while the…

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f : U \to \mathbb{R}$ be real analytic, and let $x_0 \in U$ be an isolated critical point of $f$, i.e. $\nabla f(x_0) = 0$ and…

In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely generated Kleinian group is either the whole Riemann sphere, or has measure zero.

Let $\ell^2$ denote the Hilbert space of square-summable sequences of complex numbers, with standard orthonormal basis $(e_n)_{n\in\mathbb{N}}$. Let $B$ be the C-algebra of all bou…

In harmonic analysis, a branch of mathematics, the Mizohata–Takeuchi conjecture proposed a weighted $L^{2}$ inequality for the Fourier extension operator associated with a smooth h…

Let $\mathbb{H}=\{z=x+iy\in\mathbb{C}: y>0\}$ be the upper half-plane, equipped with the hyperbolic Laplace–Beltrami operator … and let $\operatorname{SL}_2(\mathbb{Z})$ act on…

(1) Coefficients of holomorphic cusp forms. Let $k\geq 1$ be an integer and let $\Gamma\subseteq SL(2,\mathbb{Z})$ be a congruence subgroup containing…

For an integer $k\geq 2$ and a positive integer $n$, let … so that $d_{2}(n)=d(n)=\sigma_{0}(n)$ is the number of divisors of $n$, and for real $x\geq 1$ put … For…

Let $\zeta$ denote the Riemann zeta function, and assume the Riemann hypothesis, so that every non-trivial zero of $\zeta$ has the form $\tfrac12 + i\gamma$ with…

For a complex variable $s$, let … be a Dirichlet series with $a_{1}=1$ that converges absolutely for $\operatorname{Re}(s)>1$, and say that $F$ belongs to the Selberg class…

Let $F$ be a number field with ring of adeles $\mathbb{A}_F$, and let $n \ge 1$. Let $\pi = \otimes_v \pi_v$ be a cuspidal automorphic representation of $GL_n(\mathbb{A}_F)$ with u…

For $n\in\mathbb{Z}_{\geq 1}$ let $\Lambda(n)=\log p$ if $n=p^{k}$ for some prime $p$ and integer $k\geq 1$, and $\Lambda(n)=0$ otherwise, and let $\varphi$ denote the Euler totien…

Let $\zeta(s)$ denote the Riemann zeta function, i.e. the meromorphic continuation to $\mathbb{C}\setminus\{1\}$ of the function defined by $\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$ fo…

Let $\zeta$ denote the Riemann zeta function, that is, the meromorphic continuation to $\mathbb{C}$ of the function $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$, $\operatorname{Re} s1$. Cal…

In mathematics, the Hardy–Littlewood zeta function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between z…

For an open set $U\subseteq\mathbf{C}$, let $\mathcal{H}^{\infty}(U)$ denote the set of bounded analytic functions $f\colon U\to\mathbf{C}$, and for $f$ analytic near $\infty$ put…

Let $n \ge 2$ be an integer and let … be a monic polynomial all of whose roots lie in the closed unit disk, i.e. $|r_{k}|\le 1$ for every $k \in \{1,\dots ,n\}$. Since…

Let $n \ge 2$, and let $M(n)$ denote the group of rigid motions of $\mathbb{R}^n$, i.e. the maps $\sigma(x) = \rho x + b$ with $\rho \in O(n)$ and $b \in \mathbb{R}^n$. Let…

Let $d\geq 2$ be an integer, let … be a polynomial of degree $d$ over $\mathbb{C}$, and call $c\in\mathbb{C}$ a critical point of $P$ if $P'(c)=0$. Let $z\in\mathbb{C}$ be a point…

For a nonzero polynomial $P(x) = a_{0}\prod_{i=1}^{D}(x-\alpha_{i}) \in \mathbb{Z}[x]$ of degree $D$, factored over $\mathbb{C}$ with leading coefficient $a_{0}$ and roots…