9 problems

Kaneko's exponent conjecture. Hypothesis STX holds with exponents $(2,0)$. Existing results cited in the source establish weaker pairs, including $(2,1/4)$ and $(15/8,1/4)$, while…

Optimal-exponent conjecture. The optimal exponent in this error estimate is $1+\varepsilon$.

Optimal-exponent conjecture. The optimal exponent in this error estimate is $\frac{1}{2}+\varepsilon$, owing to an analogue of the Riemann hypothesis for the Selberg zeta function…

Let $\Gamma$ be the cofinite Fuchsian group defining the hyperbolic surface $\Gamma \backslash \mathbb{H}$, and let $\Psi_{\Gamma}(x)$ be its Chebyshev-like counting function. Writ…

Weil–Petersson random prime geodesic conjecture. Theorem 1.1 still holds after replacing the exponent $3/4$ by $1/2$; equivalently, there is a universal constant $c>0$ such that fo…

Let $\Gamma=\mathrm{PSL}_{2}(\mathcal{O})$, let $\lambda_j=1+t_j^2$ be the eigenvalues of the hyperbolic Laplacian on $\Gamma\backslash\mathfrak{h}^{3}$, and let $S(T,X)$ be the sp…

Let $\Gamma=SL(2,{\mathbb Z})$, and let … For $\sqrt{x}(\log x)^2\leq y\leq x$, the Brun–Titchmarsh-type estimate is $\pi_{\Gamma}(x+y)-\pi_{\Gamma}(x)\ll y$. Iwaniec's conjecture.…

The exponent conjecture. The exponent $2/3+\varepsilon$, or perhaps even $1/2+\varepsilon$, is admissible for the error term in the prime geodesic theorem; the latter exponent woul…

The quantities $X^{(d)}_{m,h}(n;\mu)$ are defined by the explicit formula preceding these observations, for the parameters and partition $\mu$ used there. Integrality conjecture. I…