Moments of C$\beta$E field partition function, $\mathsf{Sine}_{\beta}$ correlations and stochastic zeta

We prove a conjecture of Fyodorov and Keating on the supercritical moments of the partition function of the C$\beta$E field or equivalently the supercritical moments of moments of the characteristic polynomial of the C$\beta$E ensemble for general $\beta>0$ and general real moment exponents. Moreover, we give the first expression for all correlation functions of the $\mathsf{Sine}_\beta$ point process for all $\beta>0$. The main object behind both results is the Hua-Pickrell stochastic zeta function introduced by Li and Valk\'{o}.