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On irregular prime power divisors of the Bernoulli numbers. (English) Zbl 1183.11012
Summary: 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 $$| B_l/l|$$ is a product of powers of irregular primes. Let $$(p,l)$$ be an irregular pair with $$B_l/l \not\equiv B_{l+p-1}/(l+p-1) \pmod{p^2}$$. We show that for every $$r \geq 1$$ the congruence $$B_{m_r}/m_r \equiv 0 \pmod{p^r}$$ has a unique solution $$m_r$$ where $$m_r \equiv l \pmod{p-1}$$ 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.

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11R23 Iwasawa theory 11R42 Zeta functions and $$L$$-functions of number fields 11Y55 Calculation of integer sequences
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