On irregular prime power divisors of the Bernoulli numbers

Let $B_n$ ( $n = 0, 1, 2, \ldots$) denote the usual $n$th Bernoulli number. Let $l$ be a positive even integer where $l=12$ or $l \geq 16$. It is well known that the numerator of the reduced quotient $\vert B_l/l\vert$ is a product of powers of irregular primes. Let $(p,l)$ be an irregular pair with $B_l/l \not\equiv B_{l+p-1}/(l+p-1) \operatorname{mod}{p^2}$. We show that for every $r \geq 1$ the congruence $B_{m_r}/m_r \equiv 0 \operatorname{mod}{p^r}$ has a unique solution $m_r$ where $m_r \equiv l \operatorname{mod}{p-1}$ and $l \leq m_r < (p-1)p^{r-1}$. The sequence $(m_r)_{r \geq 1}$ defines a $p$-adic integer $\chi_{(p,\,l)}$ which is a zero of a certain $p$-adic zeta function $\zeta_{p,\,l}$ originally defined by T. Kubota and H. W. Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated) $p$-adic expansion of $\chi_{(p,\,l)}$ for irregular pairs $(p,l)$ with $p$ below 1000.