5 problems

Odd-moment conjecture. As $h$ tends to infinity,

Refined Hardy–Littlewood conjecture. As $N\to\infty$,

Prime-modulus rank-count conjecture. For all $n$, uniformly in $1\leq H\leq R$, and for every relevant rank threshold $r$,

Let $\mathcal H=\{h_1,\dots,h_k\}$ be a set of distinct integers, and let … be its centered singular series. The centered Hardy–Littlewood conjecture. … This is the centered form o…

Let $\delta>0$ and $Q>1/\delta$. For fixed $k$, let $S$ be the number of tuples … with $q_i\in[Q,2Q]$, $a_i$ taken modulo $q_i$, satisfying $\left\|a_i/q_i\right\|\leq\delta$ for e…