6 problems

Let $a,k,\beta\in(0,\infty)$ and let $\sigma\in(\frac{1}{2},1]$ be fixed. Mixed zeta-ratio moment conjecture. There is a constant ${\mathcal D}(\sigma,a,k,\beta)$, depending only o…

Let $\mathcal M_k(\alpha)$ be the normalized long-Dirichlet-polynomial mean square and let $\mathcal C_k(\alpha)$ be the normalized divisor variance in short intervals. Basor–Ge–Ru…

Let $k\in\mathbb N$, let $Y=Y(T)$, and let $R$ be a piecewise smooth positively oriented path encircling $0$ and $-s$. Shifted moments conjecture. For $T\geq1$, … This holds for…

For $0\leq r\leq k$, define … where $d\boldsymbol w=dw_1\cdots dw_k$, and $C_i(r)$ is the positively oriented circle of radius $1/2$ centered at $-1$ for $i\leq r$ and at $0$ other…

Let $\mathcal M_3(\alpha)$ denote the normalized mean-square quantity defined in the paper, with $\alpha>0$. The cubic shifted-moment conjecture. … The formula predicts a continuou…

Averaged additive divisor conjecture. For every fixed $k\geqslant 3$, there should exist some $\delta_k>0$ such that, uniformly for