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
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
is called a tube domain
if there is an open set uo G R
such that 0 = {sG C
: 5ft(s) G w}. We write
O = T(UJ) to denote this relation.
If R C
or C
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
3.3. Sketch of the continuation of Z(s,w) to
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,
Xd) a(s, w, d)
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)
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) -
- s,TT X Xd)a(l - 5, TT,d),
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