Asymptotic formulas for higher moments of the Hurwitz zeta function with rational parameter

Establish the conjectured asymptotic formula \(M_k(T,\alpha)\sim c_k(\alpha)T(\log T)^{k^2}\) for the moments \(M_k(T,\alpha)=\int_T^{2T}|\zeta(\tfrac12+it,\alpha)|^{2k}\,dt\) of the Hurwitz zeta function when \(\alpha\) is rational, beyond the cases \(k=1,2\) already established.

Background

For rational shifts $\alpha$, the Hurwitz zeta function decomposes into a finite linear combination of Dirichlet $L$-functions, making its moments closely related to moments of Dirichlet $L$-functions. The paper reports that an asymptotic formula analogous to the conjectured moment formula for the Riemann zeta function was proposed for $M_k(T,\alpha)$, with a constant $c_k(\alpha)$ depending on the rational parameter. The corresponding results are stated to have been established only for $k=1$ and $k=2$, leaving the conjectured formula for other moment orders unresolved.