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Selberg’s conjectures and Artin $L$-functions. (English) Zbl 0805.11062
Selberg’s conjectures describe Dirichlet series with an analytic continuation, functional equation, Euler product and a Ramanujan hypothesis. The conjectures and some of their consequences are described in J. B. Conrey and A. Ghosh [Duke Math. J. 72, 673-693 (1993; Zbl 0796.11037)]. Roughly speaking the conjectures can be viewed as an alternative to the Langlands programme, but with a more analytic flavour. The present paper strengthens this connection by showing that the Selberg conjectures imply Artin’s conjecture on the holomorphy of $L$-functions attached to nontrivial irreducible representations of finite Galois extensions. Indeed it is shown that the Langlands reciprocity conjecture also follows, for those extensions $K/k$ with $K/ℚ$ solvable.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11R42 Zeta functions and $L$-functions of global number fields 11R39 Langlands-Weil conjectures, nonabelian class field theory