The generalized Bernoulli numbers are certain algebraic numbers, defined similarly to the Bernoulli numbers, that are related to special values of Dirichlet L-functions in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.
Let χ be a primitive Dirichlet character modulo f. The generalized Bernoulli numbers attached to χ are defined by
Let ε ∈ {0, 1} be defined by χ(−1) = (−1)ε. Then,
- Bk,χ ≠ 0 if, and only if, k ≡ ε (mod 2).
Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers k ≥ 1
where L(s, χ) is the Dirichlet L-function of χ.
Read more about this topic: Bernoulli Number
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