4 problems

Asymptotic counting conjecture. The lower bound gives the true order of growth:

Let $\mathcal{A}$ be the set of positive integers $N$ for which the corresponding triple $N,N+d,N+2d$ is a three-term arithmetic progression of consecutive powerful numbers. Partit…

Let $\chi$ be a fixed Dirichlet character, set $m=\operatorname{ord}(\chi)$, and let $\theta_{\leqslant X}^{(r)}(\chi)$ count primes $p\leqslant X$ for which the corrected $\chi$-t…

For a hyperbolic $3$-manifold $M$, let $S(M,g)$ denote the number of conjugacy classes of surface subgroups in $\pi_1(M)$ of genus $g$. Kahn–Marković's conjecture. There exists a c…