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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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