11 problems

Hardy's conjecture. The infimum exponent in an estimate $P_2(r)=\mathcal{O}(r^\alpha)$ is $\frac{1}{2}$.

Gaussian-integer error-exponent conjecture. The exponent in the error term should be $\frac{1}{4}+\varepsilon$ for every $\varepsilon>0$; equivalently,

Short-interval discrepancy conjecture. For generic $\mathcal{Q}$, the corresponding short-interval error should satisfy

Generic discrepancy conjecture. For generic $\mathcal{Q}$, or for generic diagonal $\mathcal{Q}$,

Heisenberg lattice-point error exponent conjecture. For every $\epsilon>0$,

Let $N_{\sigma,\tau}(\beta,\lambda)$ denote the shifted lattice-point counting function and let $R^\gamma_{-1/2}(\beta,\lambda)$ denote the corresponding Riesz mean for the harmoni…

Let $M>0$, and consider the lattice-point problem associated with the line $f(x)=M-x$, with maximizing parameter $s>0$ and parameter $r\to\infty$. An asymptotic maximizer would be…

Let $r_k(m)$ count the integer $k$-tuples $(n_1,n_2,\ldots,n_k)$ satisfying $n_1^2+\cdots+n_k^2=m$, and let $S_k(n)=\sum_{0\leq m\leq n}r_k(m)$. Writing … where $V_k$ is the volume…

Let $\Gamma$ be a cocompact or cofinite discrete group, let $\mathcal{H}$ be a hyperbolic conjugacy class of $\Gamma$, and let $E(\mathcal{H},X;z)$ denote the corresponding error t…

Let $E(X;z,w)$ denote the error term in the classical hyperbolic lattice point problem, with $X$ the lattice-point parameter and $z,w$ points in the hyperbolic plane. Optimal error…

Gauss sphere error conjecture. The true error bound should be