180 problems

Let $P(U_1,\ldots,U_k,U_1^*,\ldots,U_k^*)$ be a noncommutative nonzero $*$-polynomial with complex coefficients, let $\underline{\lambda}\neq 0$ be a signature, and let…

Asymptotic independence conjecture. For every $x,y\in\mathbb{R}$,

DMPK limiting-density conjecture. The density $\rho(x;\mathbb{M})$ is related to $\psi(x)$ by

Log-partition function conjecture. Almost surely,

Let $C$ be a smooth projective curve of genus $g$. Assume that $C$ is in the generic family. For each $k=1,2,\dots,g$, let $a_k(m;g)$ denote the $m$-th moment of the corresponding…

Limiting-distribution conjecture. For every $a>0$ there is a Borel function $f_a:\mathbb{C}\to\mathbb{R}$ satisfying

Let $Z_X(s)$ be the truncated Hadamard factor, let $X,T\to\infty$ with $X=O((\log T)^{2-\epsilon})$, and let $k>-1/2$ be fixed. Let $G$ be the Barnes $G$-function and let $\gamma$…

Let $k$ be a positive integer, let $\theta(t)$ be the Riemann–Siegel theta function, and define Hardy's function by $Z(t)=e^{i\theta(t)}\zeta(1/2+it)$. Let $a_k$ be the arithmetic f…

Let $Z_X(s)$ denote the truncated zeta-function used in the paper, and let $A>0$ be fixed. Truncated-zeta maximum conjecture. If $2<X<\log^A T$, then, as $T\to\infty$, … The conjec…

Let $\mathcal F$ be a family of $L$-functions, and for $F\in\mathcal F$ let $c(F)$ denote its conductor. Assume the family is ordered by conductor and has approximately $D$ members…

Let $F_{\beta}(s,m)$ denote the distribution of the $m^{th}$ largest eigenvalue in the appropriate scaling limit for the Gaussian orthogonal ensemble ($\beta=1$) and Gaussian sympl…

Limiting-distribution conjecture. As $N\to\infty$ through primes, the limiting distribution of the normalized matrix elements $F_j^{(N)}$ is that of the random variable

Let $N(T)$ denote the number of non-trivial zeros $\tfrac12+{\mathrm{i}}\gamma_n$ with $0<\gamma_n\leq T$, and set … Let $F_2$ be the function specified by the paper's formula for…

Let $N(T)$ denote the number of non-trivial zeros $\tfrac12+{\mathrm{i}}\gamma_n$ with $0<\gamma_n\leq T$, and set … For $k$ with $\operatorname{Re}(k)>-1/2$, let $a(k)$ be the ari…

Let $\lambda_{\max}^{(N)}(t)$ be the largest eigenvalue of an $N\times N$ matrix undergoing Dyson's Brownian motion with parameter $\beta=1$. Let ${\cal A}_{\rm 1}$ denote the Airy…

The Riemann zeta function is defined by $[... ELLIPSIZATION...$ and continued meromorphically to the complex plane. For complex shifts $[... ELLIPSIZATION...$, define the $2k$-poin…

Let ${\bf h}^{\textrm{eff}}_i$ denote the effective channel vector for user $i$ in a system with $M$ transmit dimensions and an effective-channel subspace of dimension $N$. Chi-squ…

Let $\rho_-$ and $\rho_+$ be the left and right Bernoulli densities for the TASEP, let $\bar h(y)$ be the hydrodynamic limiting height profile, and let $h_t(j)$ be the height at ti…

Let $\widehat{\bm{S}}_n$ be the normal-scores covariance and let $\widehat{\bm{\Sigma}}^{\tau}$ be the sin-transformed Kendall matrix. Assume $p/n\to\gamma$. Spectral equivalence o…

Let $\widehat{\bm{S}}_n$ be the normal-scores covariance, and let $m_n$ and $m_F$ denote its empirical Stieltjes transform and the limiting generalized Marčenko--Pastur Stieltjes t…

Ellipsoidal separation conjecture. For every fixed $\alpha\in(0,1)$, there exists a dimension $n_0(\alpha)$ such that, for all $n\ge n_0(\alpha)$, the three ellipsoids…

Diagonal monitored-transport limit. If $L_N/N\to\tau$, then

Let $\theta>1$, let $d_0\geq 1$ be fixed, and use the spiked matrix model notation of Theorem, assuming and. Let $\bm R=\bm R^{(d)}\in\mathbb{R}^{d_0\times d}$ satisfy…

Keating–Snaith moment conjecture. For every such $\boldsymbol{\kappa}$,

Let $\mathcal{F}$ be a family of $L$-functions of type of symmetry $\mathrm{SO(even)}$ or $\mathrm{Sp}$, and let $\mathcal{F}_X$ be a finite truncation increasing to $\mathcal{F}$…