The real cuspidal moment conjecture

Let $a_{\mathrm{cusp}}(\sigma,n)$ be the cuspidal ideal-class counting function, let $\sigma\in N$ be an ideal class, and let $\beta>0$ be real. Theorem gives the order of growth of the moments under the condition in equation. Real cuspidal moment conjecture. For real $\beta>0$, Theorem still holds after removing the condition in equation; $\liminf$ and $\limsup$ in Theorem should also be equal. This predicts that the same cuspidal moment-order formula and a genuine limiting asymptotic remain valid for non-integral positive moments, despite the failure of the character condition in the displayed non-abelian example.