Kellner, Bernd C. On irregular prime power divisors of the Bernoulli numbers. (English) Zbl 1183.11012 Math. Comput. 76, No. 257, 405-441 (2007). 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. Cited in 6 Documents 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 Keywords:Bernoulli number; Riemann zeta function; \(p\)-adic zeta function; Kummer congruences; irregular prime power; irregular pair of higher order PDF BibTeX XML Cite \textit{B. C. Kellner}, Math. Comput. 76, No. 257, 405--441 (2007; Zbl 1183.11012) Full Text: DOI arXiv OpenURL Online Encyclopedia of Integer Sequences: Numerators of Bernoulli numbers B_2n. Irregular primes: primes p such that at least one of the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) is divisible by p. Irregular primes in the ratio numerator(Bernoulli(2*n)/(2*n)) / numerator(Bernoulli(2*n)/(2*n*(2*n-r))) when these numerators are different and n is a minimum for some integer r. Duplication indicates irregularity index > 1. Strong irregular primes. A prime is strong irregular iff it is a Bernoulli irregular prime and an Euler irregular prime. Irregular pairs (p,2k) ordered by increasing k. Least even integer k such that numerator(B_k) == 0 (mod 59^n). Least even integer k such that numerator(B_k) == 0 (mod 67^n). p-adic digits of the unique simple zero of the p-adic zeta-function zeta_{(p,l)} with (p,l) = (37,32). p-adic digits of the unique simple zero of the p-adic zeta-function zeta_{(p,l)} with (p,l) = (59,44). p-adic digits of the unique simple zero of the p-adic zeta-function zeta_{(p,l)} with (p,l) = (67,58). a(n) = A000367(n)/A001067(n). References: [1] J. C. Adams, Table of the values of the first sixty-two numbers of Bernoulli, J. Reine Angew. 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