5 problems

Dominance threshold conjecture. As $d\to\infty$, for every fixed $p>0.1$, the probability that the random homogeneous ideal is dominant tends to $0$.

Let $T>1$, let $\tau$ be chosen uniformly from $[T,2T]$, and let $0<\alpha<1$ with $\theta\sim(\log\log T)^{-\alpha}$. Consider an interval of size … The smoothing-scale conjecture…

Let $\operatorname{RM}$ be the random model, with a realisation determined by the shifts $a$ and factors $\rho_k(n+a)$, and let $\pi_{\operatorname{RM}}(x)$ be its counting functio…

Let ${\operatorname{RM}}_c$ denote the constrained ${\operatorname{RM}}$ described in the source, and let $\pi_{\operatorname{RM}_c}(x)$ be its prime-counting function. Let…

Typical short-interval maximum conjecture. Let $\epsilon>0$. For $T_1$ sufficiently large depending on $\epsilon$, for all $T_1\leq T\leq 2T_1$, outside a set of “bad” values of…