Keating–Snaith moment conjecture for the Riemann zeta-function

Let $\boldsymbol{\kappa}=(\kappa_{1},\kappa_{2})\in\mathbb{C}^{2}$ satisfy $\operatorname{Re}(\kappa_{1}+\kappa_{2})>-1$, and define

$$\boldsymbol{\kappa}\circ\lambda=\kappa_{1}\lambda+\kappa_{2}\overline{\lambda}$$

for $\lambda\in\mathbb{C}$. Let $G(z)$ denote the Barnes $G$-function, and define

$$a(\boldsymbol{\kappa})=\prod_{p}\left\{(1-p^{-1})^{\kappa_{1}\kappa_{2}}\sum_{m=0}^{\infty}\frac{\Gamma(\kappa_{1}+m)}{m!\,\Gamma(\kappa_{1})}\frac{\Gamma(\kappa_{2}+m)}{m!\,\Gamma(\kappa_{2})}p^{-m}\right\},$$ $$g(\boldsymbol{\kappa})=\frac{G(1+\kappa_{1})G(1+\kappa_{2})}{G(1+\kappa_{1}+\kappa_{2})}.$$

Keating–Snaith moment conjecture. For every such $\boldsymbol{\kappa}$,

$$\lim_{T\to\infty}\frac{1}{T(\log T)^{\kappa_{1}\kappa_{2}}}\int_{0}^{T}\exp\bigl(\boldsymbol{\kappa}\circ\log\zeta(1/2+it)\bigr)\,dt=a(\boldsymbol{\kappa})g(\boldsymbol{\kappa}).$$

This conjecture predicts the precise asymptotic moments of the zeta-function on the critical line, with an arithmetic factor and a random-matrix factor. It is essentially due to Keating and Snaith and remains open in this generality.