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Perspectives on the analytic theory of $$L$$-functions. Special volume of the journal Geometric and Functional Analysis. (English) Zbl 0996.11036
Alon, N. (ed.) et al., GAFA 2000. Visions in mathematics–Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25-September 3, 1999. Part II. Basel: Birkhäuser, 705-741 (2000).
This paper is an advanced survey of the modern analytic theory of $$L$$-functions. It begins with a discussion of general $$L$$-functions generated by the Langlands philosophy, and then goes on to review the major conjectures about them. This is followed by a description of function field analogues, Dirichlet $$L$$-functions, special values, subconvexity and resulting equidistribution results, $$\text{GL}(2)$$ tools, and finally, symmetry and attacks on the Generalized Riemann Hypothesis.
One of the main messages of the paper is that one should try in general to work with families of $$L$$-functions, rather than individual ones. This point is well illustrated by the description of the “amplification method”, which produces bounds for individual $$L$$-functions by averaging over an appropriate family.
For the entire collection see [Zbl 0960.00035].

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11M41 Other Dirichlet series and zeta functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses