Ideal counting function in cubic fields

Abstract

For a cubic algebraic extension K of ℚ, the behavior of the ideal counting function is considered in this paper. More precisely, let a_K(n) be the number of integral ideals of the field K with norm n, we prove an asymptotic formula for the sum {\displaystyle {\sum }_{{{\displaystyle n}}_{\mathrm{1}}^{\mathrm{2}}+{{\displaystyle n}}_{\mathrm{2}}^{\mathrm{2}}\le x}}{{\displaystyle a}}_K({{\displaystyle n}}_{\mathrm{1}}^{\mathrm{2}}+{{\displaystyle n}}_{\mathrm{2}}^{\mathrm{2}}).

Key words： Non-normal extension; ideal counting function; Rankin-Selberg convolution

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