Perspectives on the analytic theory of \(L\)-functions. Special volume of the journal Geometric and Functional Analysis. (English) Zbl 0996.11036

Iwaniec, H.; Sarnak, P.

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].

Reviewer: Roger Heath-Brown (Oxford)

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

Keywords:

generalized Riemann hypothesis; advanced survey; \(L\)-functions; Langlands philosophy; function field; Dirichlet \(L\)-functions; special values; subconvexity; equidistribution