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Zbl 1305.11093
Lee, Jun Ho
Evaluation of the Dedekind zeta functions at $s=-1$ of the simplest quartic fields.
(English)
[J] J. Number Theory 143, 24-45 (2014). ISSN 0022-314X; ISSN 1096-1658/e

The author considers cyclic quartic fields $K_t=\mathbb Q(\theta_t)$, $\theta_t$ being a zero of $X^4-tX^3-6X^2+tX+1$, where $t$ is a positive integer such that $t^2+16$ has no odd square factor (such fields are called simple quartic fields). {\it C. L. Siegel} [Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1969, 87--102 (1969; Zbl 0186.08804)] gave a formula for the value of the Dedekind zeta-function $\zeta_K(s)$ of totally real fields $K$ at negative rational integers, which for quartic fields implies $$\zeta_K(-1)={1\over30}\sum_a\sum_{I|a\delta_K}N(I),$$ where $\delta_K$ is the different of $K$, and $a$ runs over totally positive elements of $\delta_K^{-1}$ of trace $1$. The author presents a geometric method of calculating the right hand-side of this formula, and presents a table of $\zeta_{K_t}(-1)$ for $t=1,2,\dots,21$.
MSC 2000:
*11R16 Cubic and quartic extensions
11R42 Zeta functions and L-functions of global number fields
11R80 Totally real fields, etc.

Keywords: quartic fields; Dedekind zeta function; totally real field; cyclic fields

Citations: Zbl 0186.08804

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