Nonvanishing of a certain Bernoulli number and a related topic
Volume 159 / 2013
Abstract
Let $p=1+2^{e+1}q$ be an odd prime number with $q$ an odd integer. Let $\delta $ (resp. $\varphi $) be an odd (resp. even) Dirichlet character of conductor $p$ and order $2^{e+1}$ (resp. order $d_{\varphi }$ dividing $q$), and let $\psi _n$ be an even character of conductor $p^{n+1}$ and order $p^n$. We put $\chi =\delta \varphi \psi _n$, whose value is contained in $K_n=\mathbb {Q}(\zeta _{(p-1)p^n})$. It is well known that the Bernoulli number $B_{1,\chi }$ is not zero, which is shown in an analytic way. In the extreme cases $d_{\varphi }=1$ and $q$, we show, in an algebraic and elementary manner, a stronger nonvanishing result: ${\rm Tr}_{n/1}(\xi B_{1,\chi }) \not =0$ for any $p^n$th root $\xi $ of unity, where ${\rm Tr}_{n/1}$ is the trace map from $K_n$ to $K_1$.
