On irregular prime power divisors of the Bernoulli numbers

Abstract

Let $B_n$ ($n=0,1,2,...$) denote the usual $n$-th Bernoulli number. Let $l$ be a positive even integer where $l=12$ or $l\ge 16$. It is well known that the numerator of the reduced quotient $|B_l/l|$ is a product of powers of irregular primes. Let $(p,l)$ be an irregular pair with $B_l/l \not\equiv B_{l+p-1}/(l+p-1) \modp{p^2}$. We show that for every $r\ge 1$ the congruence $B_{m_r}/m_r \equiv 0 \modp{p^r}$ has a unique solution $m_r$ where $m_r \equiv l \modp{p-1}$ and $l\le m_r<(p-1)p^{r-1}$. The sequence $(m_r{)}_{r\ge 1}$ 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.

42 pages; final accepted paper, slightly revised and extended, to appear in Math. Comp

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