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 $X\ge 1$, we have ${\sum }_{n\le X}(1\star \chi )(n)=\frac{\pi }{3\sqrt{3}}X+O^{\ast }(1.94X^{1/3})$, and that, when $\chi$ is the non-principal character modulo $4$ and $X\ge 1$, we have ${\sum }_{n\le X}(1\star \chi )(n)=\frac{\pi }{4}X+O^{\ast }(1.4X^{1/3})$.