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Zbl 0964.11015
Zagier, D.
A modified Bernoulli number. (English)
[J] Nieuw Arch. Wiskd., IV. Ser. 16, No.1-2, 63-72 (1998). ISSN 0028-9825

In the note under review the author studies the rational numbers $$ B_n^*=\sum_{r=0}^n {{n+r}\choose{2r}}\frac{B_r}{n+r}, $$ where $B_n$ are the classical famous Bernoulli numbers. He shows that these modified Bernoulli numbers have some very interesting properties similar to the classical ones, namely\par a) The value of $B_n^*$ for odd $n$ is periodic mod $12$;\par b) The fractional part of $2nB_n^*-B_n$ for $n$ even is given by $\displaystyle\sum_{(p+1)|n\atop p\ \text{ prime}}\frac 1p$ ;\par c) $B_n^*\approx (-1)^{n/2}\pi Y_n(4\pi)$, $n\to\infty$, $n$ even, where $Y_n(x)$ denotes the $n$th Bessel function of the second kind.\par The proofs are really fun, as the author points out, and are examples of his virtuos handling of generating functions.
[Helmut Müller (Hamburg)]

MSC 2000:: *11B68 Bernoulli numbers, etc.

Keywords: Bernoulli numbers

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