3 problems

Let $\Lambda(n)$ be the von Mangoldt function, and for distinct integers $d_1,\ldots,d_k$ with $1\leq d_i\leq H$, let $\mathcal{D}=\{d_1,\ldots,d_k\}$. Define $E_k(x;\mathcal{D})$…

Let $\psi(x)=\sum_{n\leqslant x}\Lambda(n)$ be the Chebyshev function. Short-interval variance conjecture. For all $\varepsilon>0$, if $h\leqslant X^{1-\varepsilon}$, then … as…

Conjectured unification. The estimate