Evaluation of zeta function of the simplest cubic field at negative odd integers

Abstract

In this paper, we are interested in the evaluation of the zeta function of the simplest cubic field. We first introduce Siegel's formula for values of the zeta function of a totally real number field at negative odd integers. Next, we will develop a method of computing the sum of a divisor function for ideals, and will give a full description for a Siegel lattice of the simplest cubic field. Using these results, we will derive explicit expressions, which involve only rational integers, for values of a zeta function of the simplest cubic field. Finally, as an illustration of our method, we will give a table for zeta values for the first one hundred simplest cubic fields.