64 problems

Let $\zeta(s)$ denote the Riemann zeta function. For fixed $V$, Selberg's central limit theorem describes the distribution of $\log|\zeta(1/2+it)|$ after normalization by…

Near-optimality conjecture. This bound is conjectured to be near-optimal when $\Sigma_W$ is unrestricted.

Dunkel and Hänggi's conjecture. The limiting variance satisfies

Let $f$ be the critical circle map under consideration, and let $\mathbb T^1$ denote the circle. For an orbit starting at $x\in\mathbb T^1$, consider its effective noise under the…

Let $K\subset\mathbb{R}^n$ be a convex set of volume one, and let $X$ be uniformly distributed inside $K$. Anttila–Ball–Perissinaki conjecture. For some non-zero vector…

Let $I_{l_{1}l_{2}l_{3}}$ be the angular bispectrum, let $\Delta_{l_{1}l_{2}l_{3}}$ be its normalization factor, and let $\kappa_{4}(l_{1},l_{2},l_{3})$ denote the fourth cumulant…

Let $X_1,X_2,\ldots$ be identically distributed random variables, let $S_n$ denote their normalized sum, and let $D(S_n)$ be the squared Mallows-distance quantity used in the paper…

Let $X_1$ and $X_2$ be independent and identically distributed random variables, and let ${\mathcal{C}}(X)$ denote the Poincaré-type constant defined in the paper. Consider the nor…

Cotype-2 counterexample conjecture. If $\mathbf{B}$ is not of cotype $2$, then there exists a $\mathbf{B}$-valued random variable $X$ such that

Cotype-2 failure conjecture. If $\mathbf{B}$ is not of cotype $2$, then the equivalence between $X\in\operatorname{CLT}$ and $X\in\operatorname{CLTSS}$ is false.

Let $C_{n,n,n}$ denote the minimum cost of a perfect matching in the random $n$ by $n$ bipartite matching model, and let $\overset{\mathcal{D}}{\longrightarrow}$ denote convergence…

UCB1 allocation-variability conjecture. The large allocation variability of texttt{UCB1} contributes to its slow convergence to the central limit theorem, and the convergence rate…

Let $R_n$ be the range and $Y_n$ the stationary comparison variable defined earlier in the paper, and let $u$ be the finite quantity … Let $\kappa$ denote the variance constant fo…

Let $N_{\mathbb{R},\boldsymbol{\xi}}$ be the number of real zeros of a random polynomial from any model considered in Section. Assume that the coefficient distribution is subgaussi…

Lambert's conjecture. The $1/2$-Sobolev regularity condition is sufficient for the relevant convergence to the Gaussian distribution if and only if $\beta\le 2$.

Let ${\mathcal P}$ be the set of prime numbers, let $(X_p)_{p\in{\mathcal P}}$ be independent random variables with ${\mathbb P}(X_p=\pm1)=1/2$, and define … when $j$ is square fre…

Let $M_n$ be the length of the largest of the $n+1$ subintervals obtained after $n$ steps of Kakutani's random interval-splitting process, and let $\sigma^2=16\log 2-10$. Theorem g…

Let $\mu\in\mathcal{P}(\mathbb{R}^d)$ be an isotropic distribution, meaning that … Let $\nu_n$ denote the normalized $n$-fold convolution sequence from the central limit theorem, s…

Weighted fifth-order central limit conjecture. There exists such a weighting function $\bar w$ and a universal constant $C_3>0$ for which

Let $\alpha(n)$ be the random completely multiplicative function appearing in the paper, let $\mu$ be the Möbius function, and let $m_{x,\infty}$ be the random measure defined in t…

Let $\rho$ be a regular probability density on $\mathbb{R}^d$, let $\cov_\rho$ denote its covariance matrix, and consider the Markov kernel induced by $N$-fold convolution with…

Let $(I_n)$ be an increasing sequence of finite index sets with $|I_n|\to\infty$, and let $(X_i)_{i\in I}$ be centered random variables. Write $W_n$ for the normalized sum in the c…

Let $(I_n)$ be an increasing sequence of finite index sets with $|I_n|\to\infty$, and let $(X_i)_{i\in I}$ be centered random variables. Set … For probability measures $\nu$ and…

Let $d\geq 2$, and consider a generalized central limit theorem in $\mathbb R^d$ whose limiting law is a symmetric $\alpha$-stable probability measure with $\alpha\in(1,2)$. A rate…

For $0<\beta\leq \tfrac12$, define $n_k=\lfloor e^{k^\beta}\rfloor$ and consider the random variable sequence $(\cos(2\pi n_kx))_{k\geq 1}$ with $x$ distributed on $(0,1)$. Erdős's…