Math 254a: Dirichlet L-series
نویسنده
چکیده
1 n : (n, f) = 1 0 : otherwise , the absolute convergence for s > 1 follows from the convergence of ζQ(s) for s > 1. (ii) Given the convergence for s > 1, the proof of the product expansion is identical to the case ζQ(s). We also observe that L(s, χ1) = ζQ(s). As we will see, for non-trivial characters, the L-series behave rather differently. We will analyze these functions further, but first we want to demonstrate their relevance to us: Theorem 1.3. Given a number field K ⊂ Q(ζn), let GK be the corresponding group of Dirichlet characters modulo n. Then for s > 1 we have ζK(s) = ∏
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Math 254a: Evaluating the L-series
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