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\(L_p(1,\chi)\bmod p\). (English) Zbl 1046.11080

Let \(k\) be a number field, \(\chi\) a Dirichlet character of conductor \(\Delta\), and \(p\) an odd prime coprime to \(\Delta\). The main result of this article is the congruence \[ \chi(p)^{-1} L_p(1,\chi) \equiv \sum_{t=1}^{p-1} (1 + \tfrac12 + \ldots + \tfrac1t) (\chi(rt) + \chi(rt+1) + \ldots + \chi(rt+r-1)) \bmod p, \] where \(r\) and \(s\) are integers with \(rp + s\Delta = 1\). If \(K = \mathbb Q(\sqrt{D}\,)\) is a real quadratic number field with associated Dirichlet character \(\chi\) in which \(p\) splits, and if the right hand side of the congruence above is nonzero, then the Iwasawa \(\mu\)- and \(\lambda\)-invariant of \(K\) vanish.

MSC:

11R23 Iwasawa theory
11S40 Zeta functions and \(L\)-functions
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References:

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