Explicit count of integral ideals of an imaginary quadratic field

Abstract

We provide explicit bounds for the number of integral ideals of norms at most $X$ is $\mathbb{Q}[\sqrt{d}]$ when $d <0$ is a fundamendal discriminant with an error term of size $O(X^{1/3})$. In particular, we prove that, when $\chi$ is the non-principal character modulo $3$ and $X\ge1$, we have $\sum_{n\le X}(1\star\chi)(n) = \frac{\pi}{3\sqrt{3}}X +O^*( 1.94\,X^{1/3})$, and that, when $\chi$ is the non-principal character modulo $4$ and $X\ge1$, we have $\sum_{n\le X}(1\star\chi)(n) = \frac{\pi }{4}X+ O^*( 1.4\,X^{1/3})$.