A simple algebra A over a number field k is trivial if and only if it is everywhere
unramified, i.e. if and only if Av A 0/c kv is trivial over the completions kv of k
at all the places.
In the opposite direction, as soon as ramification is allowed, as in the case of
quaternion algebras, the internal structure of the zeta function is more complicated
and the explicit form of the global root number, the holder of ramification infor-
mation, becomes relevant. Siegel, and in a more explicit way Stark, discovered
analytic formulas (relatives of the Poisson summation formula in the additive case
and variants of the explicit formulas in the multiplicative case) capable of exhibiting
the non-triviality of the discriminant of a simple algebra over a number field. The
explicit form of these analytic approaches to the study of ramification, which uses
increasingly more information about zeros and poles of zeta functions, was devel-
oped by Stark, Odlyzko, Serre and others. A noteworthy refinement of these ideas
involves the use of the explicit formulas; for example, Mestre has obtained lower
bounds for the conductor of certain automorphic L-functions whose archimedean
components are discrete series. We give in Chapter V a development of these
ideas. The results in this chapter are best viewed as contributions to arithmetic-
algebraic geometry in the sense of Arakelov, Faltings, and Szpiro. The principal
results are The Main Formula in §4, Lemmas 2 and 3 and Theorem 6.
Chapter VI. Norbert Wiener's main contribution to analytic number theory was
his tauberian theorem which establishes the equivalence of prime number theorems
and the non-vanishing theorems for zeta functions. Whether one is interested in
the proof of a classical prime number theorem, or in one of its modern versions, e.g.
Chebotarev's density theorem, Sato-Tate distributions for the eigenvalues of Hecke
operators, analytic proofs of strong multiplicity one, the Deligne or Katz-Sarnak
monodromy distribution theorems etc., the key problem in the analytic approach
is the proof of the non-vanishing of L-functions on the boundary of the region of
absolute convergence.
The most significant progress in this direction has been the generalization by
Deligne of the method of Hadamard and de la Vallee Poussin, which estab-
lishes non-vanishing for L-functions of a wide class that includes all the classical
ones (Riemann zeta, Dirichlet L-functions, Artin-Hecke, etc.) and has the poten-
tial for further applicability to arbitrary automorphic L-functions, once a weak
version of Langlands' functoriality principle is available - analytic continuation of
the L-functions of Langlands L(s, r, n) for Re(s) 1 for all finite dimensional
representations r of the L-group, except possibly for a simple pole at s 1 when
r = 1.
There is a second method for proving non-vanishing which rests on the analytic
properties of Eisenstein series. In one version it is based on the fact that the Eisen-
stein series E(s,g) of a maximal parabolic subgroup of a connected reductive al-
gebraic group G, as a function of the complex variable s, has analytic continuation
to the unitary axis iR, and on the theory of the Whittaker functional, which
generalizes the theory of Fourier coefficients, and which for a non-trivial additive
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