28 problems

Let $c_n$ be the number of self-avoiding walks of length exactly $n$ in the square lattice $4\mathbb{Z}^2$, let $\mu_c$ be the connective constant defined by … and let $A$ be a pos…

Gath's conjecture. The optimal order of the error term is $2q-1$; that is, the infimum of the exponents $\kappa$ for which $|\mathcal E_q(t)|\lesssim t^\kappa$ holds equals $2q-1$.…

Let … be the error term in counting lattice points in a disk. Gauss circle problem conjecture. The optimal order of $P_2(x)$ is $1/4$; equivalently, for every $\epsilon>0$, … Deter…

Let $m\ge2$, and consider the theorem's lattice-point estimate for a step-two nilpotent group with a rational matrix in the sense that, for some $c>0$, its dilated matrix … is an i…

Let ${\mathbb H}^d\cong\mathbb{R}^{2d}\times\mathbb{R}$ be the Heisenberg group, and for $A=1$ and $\alpha=4$ let … The corresponding lattice-point discrepancy has a correct order…

For $n\ge 2$, let $\mathcal{B}_n(1)$ be the standard unit closed ball in $\mathbb{R}^n$, and consider the lattice-point discrepancy … as $r\to\infty$. The correct-order conjecture.…

Let $\Lambda_N^{\mathrm{sph}}$ denote the spherical truncation and let $S^{\mathrm{sph}}(N)$ be its associated incidence quantity. Spherical truncation incidence-growth conjecture.…

Selberg error-term conjecture. For every $\varepsilon>0$,

Main conjecture. For every $\varepsilon>0$,

Optimal error-term conjecture. The estimate

Lower-bound conjecture. The point count satisfies

Let $q\geq 3$ be an integer, let $N_q(x)$ count lattice points in the Heisenberg dilation of the $(2q+1)$-dimensional unit Cygan–Korányi ball, and define … where…

Götze's conjecture. The error in the lattice-point counting estimate should be

Let $n>2$, $N$ and $T$ be positive parameters, and let $\|\gamma\|$ denote the maximal absolute value of an entry of $\gamma\in\operatorname{SL}_n(\mathbb{Z})$. Consider the count…

Let $\mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\cap\delta_{x}\mathcal{B}\big|-\operatorname{vol}(\mathcal{B})x^{2q+2}$, where…

Let $\mathscr{O}=\{u\in\mathbb{R}^{2}:|u|_{2}\leq1\}$ be the Euclidean unit disk, let $uplambda_{x}u=xu$, and define the lattice-point error…

Let $varkappa_{n}=sup\big\{\alpha>0:\big|\mathcal{E}^{\mathbb{E}}_{n}(x)\big|\ll x^{n-\alpha}\big\}$, where…

Let $A$ be a sphere, and let $\boldsymbol{N}({\mathbb Z}^d,tA)$ denote the number of lattice points in $tA$. The Gauss circle problem asks for the smallest exponent $d-2+\lambda_d$…

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…

Conjectured relationship. There is a relationship between this formula for the Tutte polynomial and the algebro-geometric formula for the Tutte polynomial. The conjecture proposes…

Skriganov's conjecture. Skriganov's error term for admissible lattices is sharp.

Let $\Lambda\subseteq \mathbb{R}^d$ be a lattice, and let $B_T\subset\mathbb{R}^d$ be the ball of radius $T$ centered at the origin. Define the lattice-point counting function and…

Lattice-point remainder conjecture. It is conjectured that

Square-root error conjecture. The error term satisfies