Some time ago, Landau proved the following formula for general number fields:
$I_K(x)=U_Kx+O(x^{\delta})$, where $\delta=1-2/(1+[K:\mathbb{Q}])$, where $I_K(x)$ is the number of ideals with norm below $x$ and $U_K(x)$ is the output of the analytic class number formula.
In other words, $I_K(x)-U_Kx=O(x^{\delta})$. But what if you wanted to find the number of ideals of below norm $x$ in a given ideal class? What would the bounds be on $I_{K,C}(x)-\frac{U_K}{h_K}x$, where $I_{K,C}(x)$ is the number of ideals with norms below $x$ and $h_K$ is the number of ideal classes? Chapter 10 of Jarvis's Algebraic Number Theory shows that the ideal norms within each class have the same Dirichlet density (as part of a proof of the analytic class number formula), but that is weaker than natural density as is implied above.
