connection with the Lindelof Hypothesis, and second, because any improvement
on the convexity bound or the current best value of K tends to have dramatic
Dirichlet generalized the zeta function and introduced L-series. A well-known
example is
T( \ \rXd{n)
where \d is
character of (Z/dZ)
. These and other L-series mirror the Riemann
zeta function in that they have an analytic continuation and a functional
They also are conjectured to satisfy a corresponding generalized Riemann Hypoth-
esis. The presence of the extra parameter d leads naturally to the investigation
of the behavior of L(l/2 + it,Xd) f°r varying d, t, From this perspective, one can
formulate the Lindelof Hypothesis "in the d aspect", which states that for any e 0
there exists a constant C(e) such that for all d, |L(l/2, Xd)\ C(e)\d\e. In a manner
completely analogous to £(s) the functional equation for L(s,Xd) c a n be used to
obtain a basic convexity result: |L(l/2,x^)|
The first breaking of
convexity for L(l/2,Xd)
w a s
accomplished by Burgess [19], with K = 3/16, and
recently there has been the result of Conrey and Iwaniec [24], with K 1/6. Such
approximations to the Lindelof Hypothesis in the d aspect have important appli-
cations to such diverse fields as mathematical physics, computational complexity,
and cryptography.
The generalizations continue. One can consider, in place of ((s) or L(s,Xd),
the //-functions associated to automorphic forms on GL{r), with extra parameters
corresponding to various generalizations of Xd- In most of these instances one
expects generalizations of the Riemann and Lindelof Hypotheses to be true and the
consequences would again be remarkable.
Fortunately, if a result is elusive for a single object it is often more within reach
when the same question is asked about an average over a family of similar objects.
For example, consider the family of Dirichlet L-series L(s,Xd) with Xd quadratic
(i.e. x'd I)- This family can be collected together in the multiple Dirichlet series
* « - E ^ .
where the sum ranges over, for example, discriminants of real quadratic fields. This
is a very basic instance of the multiple Dirichlet series discussed in this article. It
is shown in [36] that Z(l/2, w) is absolutely convergent for $lw 1 and has an
analytic continuation past $lw = 1 with a pole of order 2 at the point w = 1. By
combining this result with basic Tauberian techniques one may show that there
exists a non-zero constant c such that for large X
J2 L(l/2,Xd)~cXlogX,
the sum again going over discriminants of real quadratic fields. It follows that the
average value of L(l/2,Xd) for d X takes the form of a constant times logX
In fact, Euler had used his theory of divergent series to guess the functional equation of the
zeta function roughly a century before Dirichlet and Riemann. It is also interesting to note that
Dirichlet L-functions were defined before the zeta function was studied by Riemann as a function
of a complex variable. We thank the referee for calling these historical facts to our attention.
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