12 GAUTAM CHINTA, SOLOMON FRIEDBERG, AND JEFFREY HOFFSTEIN

The situation for GL(1), GL(2) and GL(3) is different. There we can m,ake

the heuristic rigorous and thereby prove continuation to

C2

without using Rankin-

Selberg integrals! Applications (non-vanishing, mean-value theorems) then follow.

The key point is to take advantage of the finite group of functional equations, and

Hartogs' Continuation Principle.

3.2. Hartogs' continuation principle and Bochner's tube theorem.

To overcome the obstructions that arise in the Rankin-Selberg integral method of

studying multiple Dirichlet series, we shall employ Hartogs' Principle in a stronger

form due to Bochner. Let us describe this now. We recall the definition:

DEFINITION 3.1 (Tube Domain). An open set Q c C

m

is called a tube domain

if there is an open set uo G R

m

such that 0 = {sG C

m

: 5ft(s) G w}. We write

O = T(UJ) to denote this relation.

If R C

Mm

or C

m

and m 2, let R be the convex hull of R. It is easy to see

that if Q — T(uS) then ft = T(Co). Then the relevant result is

THEOREM 3.2. (see Hormander [41], Theorem 2.5.10) If ft is a connected tube

domain, then any holomorphic function in Q can be extended to a holomorphic

function on Cl.

Thus if we can continue a meromorphic function whose polar divisor is a finite

number of hyperplanes to Q it automatically extends meromorphically to Cl, since

its product with a finite number of linear factors is holomorphic. In many cases

this is exactly what occurs with multiple Dirichlet series.

The theorem above is due to Bochner. A weaker result of Hartogs states that

there are no compact holes in domains of holomorphy in more than one complex

variable.

3.3. Sketch of the continuation of Z(s,w) to

C2

for GL(r) for r

3. We can now sketch the continuation of Z(s, w) for n on GL(r) with r 3.

First, suppose that we can introduce some weight functions a(s, 7r, d) so that the

interchange of summation is actually valid. The original heuristic interchange of

summation implicitly assumed everything was square-free, which is not the case.

We assume now that with appropriate weight factors this interchange will in fact

work. The weight factors do exist; see Sections 3.4, 3.5 below for more details.

Moreover, as we shall explain there, they are unique—for r 3 there are unique

factors that allow the sum of Euler products in s to equal a sum of Euler products

in w [17]!

The relevant series to look at is

(3.7) Z(s, w) = Y,

L(s

*

x

Xd) a(s, w, d)

C(d)\d\'u',

where £ is on GL(1) and TT is an automorphic form on GL(r) with r 3. When

the weight factors a(s, 7r, d), b(w, £, 7r, m) are chosen correctly, this can be rewritten

after applying quadratic reciprocity as

(3.8) Z(s, w) = ] T L(w, £Xm) b(w, £, TT, m)

\m\~s.

In addition to allowing this interchange of summation, the weighting factors, when

multiplied by the L-functions, satisfy the functional equations

(3.9) L(s,TT x \d)a(s,TT, d) -

\d\r{^~s)L(l

- s,TT X Xd)a(l - 5, TT,d),