Editing Bernoulli number (section)

==<span id="Generalized Bernoulli numbers"></span>Generalized Bernoulli numbers==
The '''generalized Bernoulli numbers''' are certain [[algebraic number]]s, defined similarly to the Bernoulli numbers, that are related to [[Special values of L-functions|special values]] of [[Dirichlet L-function|Dirichlet {{mvar|L}}-functions]] in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.

Let {{mvar|χ}} be a [[Dirichlet character]] modulo {{mvar|f}}. The generalized Bernoulli numbers attached to {{mvar|χ}} are defined by

: <math>\sum_{a=1}^f \chi(a) \frac{te^{at}}{e^{ft}-1} = \sum_{k=0}^\infty B_{k,\chi}\frac{t^k}{k!}.</math>

Apart from the exceptional {{math|''B''<sub>1,1</sub> {{=}} {{sfrac|1|2}}}}, we have, for any Dirichlet character {{mvar|χ}}, that {{math|''B''<sub>''k'',''χ''</sub> {{=}} 0}} if {{math|''χ''(−1) ≠ (−1)<sup>''k''</sup>}}.

Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers {{math|''k'' ≥ 1}}:

: <math>L(1-k,\chi)=-\frac{B_{k,\chi}}k,</math>

where {{math|''L''(''s'',''χ'')}} is the Dirichlet {{mvar|L}}-function of {{mvar|χ}}.{{r|Neukirch1999_VII2}}

===Eisenstein–Kronecker number===
{{main|Eisenstein–Kronecker number}}
[[Eisenstein–Kronecker number]]s are an analogue of the generalized Bernoulli numbers for [[imaginary quadratic field]]s.{{r|Charollois-Sczech|BK}} They are related to critical ''L''-values of [[Hecke character]]s.{{r|BK}}