5 problems

Lower-bound conjecture. The point count satisfies

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 $\mathbb{T}=\mathbb{R}^2/\mathbb{Z}^2$ be the rational two-dimensional flat torus, and let $R_{\mathbb{T}}(T)$ denote the error term in Weyl's law for its eigenvalues. Hardy's…

On the line $\operatorname{Re} s=\frac{1}{2}$, the relevant Laplace transform has infinitely many poles at $s=\frac{1}{2}\pm it_j$ arising from the discrete spectrum, with correspo…

Let $F(x)$ be defined by … where $[x]$ is the greatest integer less than or equal to $x$. Ramanujan's sine-series conjecture. If $\theta\in(0,1)$ and $x>0$, then … This is a claime…