4 GAUTAM CHINTA, SOLOMON FRIEDBERG, AND JEFFREY HOFFSTEIN

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

consequences.

Dirichlet generalized the zeta function and introduced L-series. A well-known

example is

T( \ \rXd{n)

n=l

where \d is

a

character of (Z/dZ)

x

. These and other L-series mirror the Riemann

zeta function in that they have an analytic continuation and a functional

equation1.

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^)|

C(e)|d|4+e.

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 ^ .

d

a

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,

0dX

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.