23 problems

Let $T_k^n$ denote the relevant point-to-point passage time, and let $\operatorname{Var}(T_k^n)$ be its variance. Variance conjecture. The variance satisfies … This prediction is u…

Friedlander–Goldston conjecture. The Hooley asymptotic $G(X,Q)\sim X\log Q$ should hold when $X^{1/2+\epsilon}<Q<X$, and, when $X^{1/2+\epsilon}<Q<X^{1-\epsilon}$,

Let $N_{I,\boldsymbol{\xi}}$ denote the number of real zeros in a natural interval $I$ for one of the random polynomial models considered in the paper. Suppose that the coefficient…

Let $X_1$ and $X_2$ be independent quantum computing errors in an $n$-qubit computation, with variances $V_1(X_1)$ and $V_2(X_2)$, respectively. Variance formula conjecture. The va…

Least $p$-variances conjecture.

Let $L_n^{\varoast}$ denote the relevant longest monotone subsequence length for $\varoast\in\{\boxslash,\boxbslash,\boxdot\}$, and let $\gamma(\varoast)$ be the corresponding scal…

Variance asymptotic conjecture. The variance of the count of squarefull numbers in these intervals satisfies

Unified variance-scale conjecture. Define

Four-regime variance conjecture. The following asymptotic estimates hold: (i) if $x\to0$, then

Best-case variance conjecture. One has

Let $D$ be a squarefree fundamental discriminant. For $0\leq r<R$, let $A_{r,R}(w)$ be the hyperbolic annulus centered at $w$ with inner radius $r$ and outer radius $R$, and let…

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Let $X\in\{0,\ldots,n\}$ be a random variable with $p_0p_n>0$ and probability generating function $f_X$. Let $\delta>0$ and $R\geqslant1$. If every zero $\zeta$ of $f_X$ satisfies…

Let $d_z(n)$ be the generalized divisor function and define the arithmetic-progression variance by … Let $a_z$ and $\alpha_1^{(z)}$ be as in the paper. The arithmetic-progression d…

For $z>0$, let $d_z(n)$ be the generalized divisor function, define $D_z(x)=\sum_{n\leq x}d_z(n)$, and let $\overline D_z(x)$ be the contour approximation defined in the paper. Set…

Let $V_b(X;H)$ be the variance defined from the error in the smooth approximation to counts of sums of two squares in intervals of length $H$. The order-of-magnitude variance conje…

Let $b(n)$ be the indicator of integers representable as sums of two squares. Let $B(x)=\sum_{n\leq x}b(n)$, let $I(x;H)$ be the paper's smooth approximation to $B(x+H)-B(x)$, and…

Higher-dimensional excess-variance conjecture. When $d\geq3$, the variance of this count as the dilation parameter ranges over $0<r<R$ should be much larger than $|\Omega(R,t)|$. T…

Let $\Omega(r,t)$ denote the thin annular region used in the paper, and write the main contribution to the variance as $X(r,t)+Y(r,t)$, where $Y(r,t)$ is the series containing the…

High-variance network conjecture. For the final estimate to have large variance, some upstream agents should have a disproportionately large out-degree, while the remaining agents…

Sublinear variance conjecture. The estimate $$ is satisfied for all Cayley graphs of $G$.

Let $P_{n,d}$ be the graph on $\\{0,1,\ldots,n\\}$ with edges between vertices of different parity whose distance is at most $2d+1$. Let $\operatorname{Lip}(P_{n,d},0)$ be the set…

Let $a_n(h)$ denote the first-passage percolation time across a cylinder of length $n$ and half-height $h$, and let $d\ge 2$. Thin-cylinder variance-order conjecture. The variance…