An analogue of Kummer's relation between the ideal class number and the unit index of cyclotomic fields

Abstract

In this paper, we obtain a formula for the special value of Euler-Dirichlet $L$-function $L_E(s,\chi )$ at $s=1$. This leads to another class number formula of $\mathbb{Q}({\mu }_m{)}^{+}$, the maximal real subfield of $m$th cyclotomic field. From this formula, we construct a new type of cyclotomic units in $\mathbb{Q}({\mu }_{p^n})$, which implies a similar Kummer's relation between the ideal class number of $\mathbb{Q}({\mu }_{p^n}{)}^{+}$ and the unit index.

13 pages