Nonvanishing of a certain Bernoulli number and a related topic

Humio Ichimura

Humio Ichimura

Acta Arithmetica, Tome 161 (2013), p. 375-386 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $p = 1 + 2^{e + 1} q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e + 1}$ (resp. order $d_{φ}$ dividing q), and let ψₙ be an even character of conductor $p^{n + 1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $K ₙ = ℚ (ζ_{(p - 1) p ⁿ})$ . It is well known that the Bernoulli number $B_{1, χ}$ is not zero, which is shown in an analytic way. In the extreme cases $d_{φ} = 1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T r_{n / 1} (ξ B_{1, χ}) \neq 0$ for any pⁿth root ξ of unity, where $T r_{n / 1}$ is the trace map from Kₙ to K₁.

Publié le : 2013-01-01

Zbl 1284.11143

EUDML-ID : urn:eudml:doc:279585

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     author = {Humio Ichimura},
     title = {Nonvanishing of a certain Bernoulli number and a related topic},
     journal = {Acta Arithmetica},
     volume = {161},
     year = {2013},
     pages = {375-386},
     zbl = {1284.11143},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-4-6}
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Humio Ichimura. Nonvanishing of a certain Bernoulli number and a related topic. Acta Arithmetica, Tome 161 (2013) pp. 375-386. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-4-6/