MULTIPLE DIRICHLET SERIES AND AUTOMORPHIC FORMS 23

To study a sum of twists via Hartogs' principle/Bochner's theorem, the relationship

between e(s, 7r) and e(s, n x Xd ) *s needed. The quotient is a power of the conduc-

tor, which is essentially the square-free part of n, times the quotient at s — 1/2.

This last factor is essentially the r-th power of a Gauss sum of order n, G(xYi that

has been normalized to have absolute value 1. (More precisely, this is true after

sieving, so it is more convenient to work with a finite dimensional vector space

of multiple Dirichlet series; see Fisher-Friedberg [31] for a discussion of this point

in a classical language and Brubaker-Bump [7] for a discussion which is adelic in

nature.)

Because of this crucial change, the heuristic that describes the quadratic twist

case is not useful. In fact, after a functional equation one obtains a new mutliple

Dirichlet series—not Z(s,w), but a series whose weight factors involve n-th order

Gauss sums. A similar situation occurs if one interchanges and then applies a func-

tional equation. Moreover, these two operations need not commute (even ignoring

scattering matrix and bad prime considerations)! To use the convexity methods of

Section 3.2, one is then led to consider several different families of multiple Dirichlet

series that are linked by functional equations. We discuss two cases in detail (n-fold

twists of GL(1) and cubic twists of GL(2). This is followed by a discussion of the

nonvanishing of n-th order twists of a GL(2) automorphic L-function for arbitrary

n. Though the sum of twisted L-functions Z(s, w) has not been continued to

C2,

a

variation on the method of double Dirichlet series gives an interesting result.

4.1. n-Fold Twists of GL(1). The study of the sum of the n-fold twists of a

given Hecke character was carried out by Friedberg, Hoffstein and Lieman [34]. One

obtains two different families of multiple Dirichlet series: the n-th order twists of the

original L-function ^2 L(s, £x^ )a(s, £,

f)|i|-ti;

and a multiple Dirichlet series built

up from infinite sums of n-th order Gauss sums. The second series is obtained from

the first by use of the functional equation for L(s, %Xd

f°ll°wed

by an interchange

of summation. But these latter sums arise as the Fourier coefficients of Eisenstein

series on the n-fold cover of GL(2), and they can thus be controlled by using the

theory of metaplectic Eisenstein series. In particular, they satisfy a functional

equation of their own, even though they are not Eulerian! To keep this paper to

manageable length, we do not give many details; we will supply them in the more

complicated case of GL(2) below. We remark that automorphic methods, which

could be for the most part avoided in the quadratic twist case, seem unavoidable

in many problems involving n-th order twists for n 2.

In the case at hand, the continuation of the two families of double Dirichlet

series to

C2

is established from Bochner's theorem. Note that earlier we mentioned

that such a sum could be approached by an integral of an Eisenstein series on

the n-fold cover of GL(n). Thus the Hartogs/Bochner-based method allows one to

replace the use of Eisenstein series on the n-fold cover of GL(n) with the use of

Eisenstein series on the n-fold cover of GL(2), which are considerably simpler. We

shall see a similar reduction to GL(2) in the work on Weyl group multiple Dirichlet

series that is discussed in [9].

Let us also note that Brubaker and Bump ([8], in this volume) have obtained the

double Dirichlet series discussed in this section as residues of Weyl group multiple

Dirichlet series, and have shown that their functional equations may be understood