6 problems

For positive integers $a,b$, let $LQ_{a,b}$ be the quaternion ring … and let $LQ_{a,b}^k$ denote the additive group generated by all $k$th powers in $LQ_{a,b}$. Quaternion-ring War…

Let $k\geq 2$, let $s$ be a positive integer, and define … where $e(\theta)=\exp(2\pi i\theta)$ and $X\geq 1$. The main conjecture in Vinogradov's mean value theorem. For every…

Critical mean-value conjecture. For every $p\geq d+1$ and every $\varepsilon>0$, there is a constant $C_{d,p,\varepsilon}>0$ such that

Weak Weyl-sum conjecture. There is a constant $C_{d,\varepsilon}>0$ such that, for every $\varepsilon>0$,

Sharp Weyl-sum conjecture. For every $\varepsilon>0$, there is a constant $C_{d,\varepsilon}>0$ such that

Let $N$ be a sufficiently large integer, and let $f_{1},\ldots,f_{r}$ be numbers of the specified type for the representation problem under consideration. Generalized Waring–Goldba…