INTRODUCTION xvii

"natural operator problem" whose solution has a well defined trace given by both

sides of the explicit formulas. An answer to this problem will certainly be accom-

panied with new insights into the dual relation between the primes and zeros of

as).

There is no doubt that the explicit formulas have proven their worth in the

study of the distribution of prime numbers. They have also been very suggestive

in the study of group representations, as already noted by Selberg 2 in reference to

the trace formula for SL(2):

"... it has a rather striking analogy to certain formulas that

arise in analytic number theory from the zeta and L-functions of

algebraic number fields."

An observation that lead him to the introduction of the Selberg zeta function.

Given the new work of Connes, which adds significance to the general notion

of trace, we expect that the explicit formulas will continue to exercise an ever

increasing role in the study of L-functions.

We remark that in the meantime it is possible, using an observation of Delsarte,

to calculate certain invariants of operators, e.g. the regularized expression

exp(^]log p),

p

an expression that attaches a sense to the notion of determinant associated to

zeta and L-functions. Another important application of the explicit formulas, not

considered in this book, is Montgomery's work on the Pair Correlation Hypothesis,

a deep insight on how the zeros of £(s) are jointly distributed. The work of Sarnak

and Rudnick has demonstrated that this is a very general phenomenon that applies

to the larger class of automorphic L-functions on GL(n). The explicit formulas of

Weil are studied in Chapter IV. The main results are Theorem 4A and Theorem

4B.

Chapter V. The diophantine properties of number fields are intimately connected

with the presence of ramification as was early realized by Kronecker. The well

known theorem of Minkowski:

\disc(k)\l, k^Q,

has a representation theoretic interpretation, as follows from a result of Hamburger,

to the effect that the trivial representation is the only Galois representation

r :

Gal(Q8ep/Q)

— GL

n

(C),

which is unramified everywhere, including at the archimedean primes. Artin noted

that the zeta functions of simple algebras over number fields could be calculated

explicitly, and that their precise form, including the exact location of their poles,

could be used advantageously to prove the central theorem of class field theory:

See A. Selberg, "Harmonic Analysis and Discontinuous Groups ...", Indian Jour., p. 75.