Note on Bernoulli numbers associated to some Dirichlet character of prime conductor

Abstract

We fix an integer ${n \geq 1}$ and a divisor m of n such that n/m is odd. Let p be a prime number of the form ${p=2n\ell+1}$ for some odd prime number ${\ell}$ with ${\ell \nmid m}$. Let ${S=pB_{1,2m\ell}}$ be the p times of the generalised Bernoulli number associated to an odd Dirichlet character of conductor p and order ${2m\ell}$, which is an algebraic integer of the ${2m\ell}$th cyclotomic field. It is known that ${S \neq 0}$. More strongly, we show that when ${\ell}$ is sufficiently large, the trace of ${\zeta^{-1}S}$ to the ${2m}$th cyclotomic field does not vanish for any ${\ell}$th root ${\zeta}$ of unity. We also show a related result on indivisibility of relative class numbers.