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Mathematics of Computation

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On irregular prime power divisors of the Bernoulli numbers

Author: Bernd C. Kellner
Journal: Math. Comp. 76 (2007), 405-441
MSC (2000): Primary 11B68; Secondary 11M06, 11R23
Published electronically: August 1, 2006
MathSciNet review: 2261029
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Abstract: 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.

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Additional Information

Bernd C. Kellner
Affiliation: Mathematisches Institut, Universität Göttingen, Bunsenstr. 3–5, 37073 Göttingen, Germany

Keywords: Bernoulli number, Riemann zeta function, $p$-adic zeta function, Kummer congruences, irregular prime power, irregular pair of higher order
Received by editor(s): June 15, 2005
Received by editor(s) in revised form: September 4, 2005
Published electronically: August 1, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.