IAS/Park City Mathematics Series
Volume 12, 2002Analytic Number Theory and Families of
Automorphic £-functions
Philippe Michel
Foreword
These notes describe several applications of methods from analytic number the-
ory to the theory of automorphic £-functions with a special focus on the methods
involving families. Families of £-functions occur naturally in Analytic Number The-
ory. Indeed, many arithmetical quantities can be evaluated via harmonic analysis
on some appropriate space of automorphic forms: the original object, <p say, is de-
composed spectrally into a sum of eigencomponents along a family :F = { 7f} of au-
tomorphic representations. Then a "simple" method to evaluate the original object
is to combine a trivial averaging over :F with non-trivial bounds for each eigencom-
ponent <p"; in many instances the bounds can be deduced from the (non-trivial)
analytic properties of an £-function attached to each individual 7f, L(7r, s) say. A
typical example following this principle is the problem of counting the number of
primes in a given arithmetic progression: the standard method, which goes back
to Dirichlet, is to express the characteristic function of the arithmetic progression
as a sum over the family :F = { x( mod q)} of characters with the given modulus.
Then the possibility of accurately counting the primes in this progression depends
directly on the existence of a non-trivial zero free region for the £-function of each
character; moreover, the quality of the counting is linked directly to the width of
the zero-free region. Of course, establishing the individual analytic properties of£-
functions is one of the main goals of Analytic Number Theory; however, in these lec-
tures we will not focus (directly) on the individual aspects but rather on the global
properties of the underlying families of £-functions {L(7r, s) }nEF· A first motivation
is that for most advanced problems, the above, rather direct, method may not be
Lecture l. Analytic properties of individual £-functions
not available unconditionally (like the Generalized Riemann Hypothesis) or, even
if they were available, they wouldn't be strong enough (even the Generalized Rie-
mann Hypothesis has some limitations!). A good part of the analytic number theory
(^1) Mathematiques, Universite Montpellier II CC 051 , 34095 Montpellier Cedex 05, FRANCE ..
E-mail address: michel©math. uni v-montp2. fr.
The author was partially supported by the Institut Universitaire de France and the ACI Jeunes
Chercheurs "Arithmenque des fonctions L".
@2 007 American Mathematical Society
181