analytic number theory declares that the central questions will remain open until
we have gained a better understanding of the nature of all the primes including
the archimedean ones. It is in this spirit that the verification of the integrality
of L-functions remains a difficult problem intimately related to the presence of
gamma factors. A similar instance is the location and distribution of the zeros of
L-functions, a problem which rests ultimately on bounds for the local archimedeam
L-factors. An important result in this connection is Riemann's product formula:
(Product Formula)
J] ^ r = ji^fi IIU " ^
^ ' p "
where the product on the right-hand side is taken over all the zeros of £(s) in the
strip 0 Re(s) 1, it being understood that p and 1 p go together.
The study in depth of the archimedean primes and their local L-factors starts
with the well known characterization theorem of Bohr-Mollerup:
The Euler gamma function T(s) is the only function defined for real s 0, which
is positive, is 1 at s=l, satisfies the functional equation sT(s)=T(s + 1), and is
logarithmically convex, that is, log T(s) is a convex function on the real line.
The uniqueness of the multiplication formula for the gamma function, viewed as an
identity between local L-factors, is the arithmetic manifestation of the basic fact
that the real archimedean prime admits a unique non-trivial extension: the complex
one. The analytic properties of the gamma function stem from the realization that
log |r(s)| = Re logr(s) is a harmonic function, a fact already used by Rademacher
(and to some extent also by Siegel) to obtain strong Phragmen-Lindelof estimates
for r(s). This circle of ideas have applications to growth estimates for L-functions
with at most a finite number of poles.
The above remarks set the tone for the emphasis given to number fields in this
book. Chapter III is an exposition of these ideas, classically known as convexity
estimates. It includes estimates for the automorphic L-functions of principal type
on GL(n), as well as for the Artin-Hecke L-functions. The main results in this
chapter are Riemann's Formula in Theorem 3 and the estimates in Theorems 14A,
14B, and 14C.
Chapter IV. The linearization of Riemann's product formula, obtained by differ-
entiating the logarithms of both sides, leads to explicit relations between the primes
(finite and archimedean) and the non-trivial zeros of ((s). These relations, when
integrated against particular functions $(s) over suitable domains in the s-plane
provide new identities that form the core of many applications of zeta functions to
questions about number fields. The internal symmetries possessed by these formu-
las seemed to have been first observed by Riemann himself, but were not developed
to any great extent until the middle of the last century when Guinand, then Del-
sarte, and finally Weil found generalizations which exhibited some kind of duality
between zeros and primes. In its modern formulation, Weil's explicit formula is an
equality between two distributions, one associated to the zeros of an L-function,
and another - its "Fourier Dual"- associated to the primes. It has been a goal of
many number theorists, a goal which remains unfulfilled to this day, to formulate a
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