Hybrid Euler–Hadamard independence conjecture for ζ′ moments

Let $\rho$ range over the non-trivial zeros of $\zeta(s)$, let $N(T)$ count zeros with $0<\operatorname{Im}(\rho)\leq T$, and let $P_X(s)$ and $Z_X(s)$ denote the Euler and zero factors in the paper's hybrid Euler–Hadamard product for $\zeta(s)$. Suppose $\varepsilon>0$ and $X,T\to\infty$ with $X=O((\log T)^{2-\varepsilon})$. Hybrid independence conjecture. For any $k>-3/2$,

$$\frac{1}{N(T)}\sum_{0<\gamma\leq T}|\zeta'(\rho)|^{2k}\sim\left(\frac{1}{N(T)}\sum_{0<\gamma\leq T}|P_X(\rho)|^{2k}\right)\left(\frac{1}{N(T)}\sum_{0<\gamma\leq T}|Z_X'(\rho)|^{2k}\right).$$

This asserts asymptotic independence of the Euler-product and zero-factor contributions to the derivative moments. The paper proves supporting cases $k=1$ and gives additional evidence for $k=2$, but the general assertion is open.