Nonvanishing of a certain Bernoulli number and a related topic

Humio Ichimura¹

¹ Faculty of Science Ibaraki University Bunkyo 2-1-1 Mito, 310-8512, Japan

Acta Arithmetica, Tome 159 (2013) no. 4, pp. 375-386.

Résumé

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 $K_1$.

Détail

DOI : 10.4064/aa159-4-6

Keywords: odd prime number odd integer delta resp varphi odd resp even dirichlet character conductor order resp order varphi dividing psi even character conductor order put chi delta varphi psi whose value contained mathbb zeta p known bernoulli number chi zero which shown analytic extreme cases varphi algebraic elementary manner stronger nonvanishing result chi nth root unity where trace map