5 problems

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…

Typical finite ring conjecture. The fraction of isomorphism classes of commutative rings $A$ of order at most $N$ satisfying all of the following conditions tends to $1$:

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

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…

Let $H$ be a simple Lie group of real rank at least $2$. Let $x$ be a positive real number, and let $\gamma(H)$ be the explicit constant computable from the absolute root system of…