Bernoulli numbers and zeros of \(p\)-adic \(L\)-functions (Q1038638)

From the introduction: For a prime \(p\) and for a nonprincipal even Dirichlet character \(\chi\) whose conductor is not divisible by \(p^2\) (or by 8, if \(p=2\)), consider the Leopoldt-Kubota \(p\)-adic \(L\)-function \(L_p(s,\chi )\). Let \(\theta\) denote the \(p\)-free component of \(\chi\). The aim of the present article is to study a relationship between rational \(p\)-adic zeros of \(L_p(s,\chi )\) and the \(p\)-divisibility of the Bernoulli numbers \(B^m(\theta )\) as \(m\) tends to infinity. As is to be expected, this relationship depends on the Iwasawa \(\lambda\)-invariant attached to \(\chi\)'.