Conjecture on moments of the derivative of the Riemann zeta-function

Let $k$ be a positive integer, let $\theta(t)$ be the Riemann–Siegel theta function, and define Hardy's function by $Z(t)=e^{i\theta(t)}\zeta(1/2+it)$. Let $a_k$ be the arithmetic factor

$$a_k=bprod_p\left(1-\tfrac{1}{p}\right)^{k^2}bsum_{m=0}^{\infty}\left(\frac{bGamma(m+k)}{m!\,bGamma(k)}\right)^2p^{-m}.$$

Let $b_k$ and $b_k'$ be the constants defined by the corresponding characteristic-polynomial moment asymptotics. Zeta-derivative moment conjecture. As $T\to\infty$,

$$\frac{1}{T}\int_0^T|\zeta'(1/2+it)|^{2k}\,dt\sim a_kb_k\log(T)^{k^2+2k},$$

and, similarly,

$$\frac{1}{T}\int_0^T|Z'(t)|^{2k}\,dt\sim a_kb_k'\log(T)^{k^2+2k}.$$

These predictions are obtained from the random-matrix model for the Riemann zeta-function and its derivative. The paper establishes the corresponding asymptotics for derivatives of characteristic polynomials of random unitary matrices, while the zeta-function and Hardy-function moment asymptotics remain conjectural.