20 problems

Let $\mathcal{R}$ be the family of random variables of the form $X=\sum_{k=1}^n a_k\varepsilon_k$, where $n\geq 1$, $a_k\in\mathbb{R}$, $\sum_{k=1}^n a_k^2=1$, and the…

Eaton's conjecture. For $x>\sqrt{2}$,

Hitczenko–Kwapień conjecture. $C=7/32$, with equality attained by the example $a_1=\dots=a_6$.

Equal-weights conjecture. For every $x$, the supremum in the definition of $F(x)$ is attained by a Rademacher sum having all of its weights equal.

For each $g\in M_{24}$, let $n_g$ be the order of $g$, let $h_g$ be the length of a minimal cycle in its induced permutation, and let $H_g$ denote the corresponding McKay–Thompson…

Conjecture for $F$ on $[\sqrt{2},2]$. For $x\in[\sqrt{2},2]$,

Lowther's conjecture. The function $f$ is piecewise constant on $\mathbb{R}_{\geq 0}$, takes seven different values, and has discontinuities at

Universal Rademacher-Gaussian comparison conjecture. The comparison in (R-G) continues to hold with a universal constant for every $d$ and every $n$ vectors in $\mathbb{R}^d$.

Let $X=\sum_{i=1}^n a_i\varepsilon_i$ be a Rademacher sum, where the $\varepsilon_i$ are independent uniformly random signs taking values in $\{-1,1\}$, and suppose that…

Let $x_1,\ldots,x_n$ be independent random variables uniformly distributed in $\{-1,1\}$, and let … be a Rademacher sum with $\operatorname{Var}(X)=1$ and $0<a_i\leq a_1$ for every…

Let $N>1$ be an integer, and let $R_{N,1}(n)$ denote the Rademacher-sum sequence associated with level $N$. The genus $g_0(N)$ is the genus of the modular curve corresponding to…

Let $X$ be a Niemeier root system and let $g\in G^X$. Let $\check{H}^X_g$ denote the vector formed by the components of $H^X_g$ indexed by $0<r<m$, and let…

Let $\Gamma_g$ be the group associated to $g\in\mathbb{M}$, and let $T^{(-1)}_{\Gamma_g}$ denote the normalized Rademacher sum attached to $\Gamma_g$. Duncan--Frenkel conjecture. T…

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent identically distributed Rademacher random variables, with $\mathbb{P}\{\varepsilon_i=\pm1\}=1/2$, and let … where…

For $\ell\in\{2,3,4,5,7,13\}$ and $g\in G^{(\ell)}$, let $H^{(\ell)}_g$ be the umbral McKay–Thompson series, and let $R^{[\mu]}_{\Gamma_0(n_g),\psi^{(\ell)}\rho^{(\ell)}_g,1/2}$ de…

Let $S_n$ be the normalized sum of independent Rademacher variables, and let $Z$ be a standard normal random variable. Efron's conjecture. There exists an upper bound for…

Let $\mathfrak{G}$ be the set of subgroups of $G({\mathbb Q})$ commensurable with $G({\mathbb Z})$, let $\mathfrak{I}_0=\bigcup_{\Gamma\in\mathfrak{G}}I_0(\Gamma)$, and let…

Let $G$ be the group acting on the upper half-plane, let $\mathfrak{G}$ be the set of subgroups of $G({\mathbb Q})$ commensurable with $G({\mathbb Z})$, and let…

The conjecture. The lower-tail estimate

The conjecture. The estimate holds with constant $c=\frac12$. The source records the currently stated real-valued bound as $\frac38$ and calls the one-half assertion well known.