MULTIPLE DIRICHLET SERIES AND AUTOMORPHIC FORMS
7
extension F( yd)/ F. Thus the sum of the numbers L(s, n x \d) is an infinite sum of
one-variable Dirichlet series—a multiple Dirichlet series. More generally, one may
introduce a weighting factor a(s,7r, d) and construct
(2.1) Z(s,w) =
_ y L(s,7r x Xd)a(s^,d)
V
\d\w
Such a series will converge for 5ft(s), $l(w) sufficiently large. The numerators are
Langlands L-functions on GL(r) and so each continues individually to all complex
s. Our goal is to find appropriate weighting factors a(s,7r, d) so that this series is
well-behaved in w. Indeed, as we shall explain, in some cases weight factors exist
such that the double Dirichlet series (2.1) possesses meromorphic continuation to
all (s,w) G
C2
and moreover satisfies a finite group (typically non-abelian) of
functional equations in (s,w).
In the case that L(s,ir) is a product of lower rank L-functions at shifted argu-
ments L(s, 7r) = n[=i L(si ^i)? Z{s- w) ls a multiple Dirichlet series of the form
7(
x y ^ (n[=i L0* *i x Xd)) a({si}i {^}, d)
Z(sus2,-' ,sr,w) = 2 ^ |^p
for suitable weight factors a. One may study these series by similar methods.
2.2. A first example. Why is a series such as (2.1) a reasonable thing to
construct? We begin with the case of GL(1). Let ^(7, z) be the theta multiplier
where e^ = 1 if d = 1 mod 4, e^ = i if d = 3 mod 4, (^) is a (quadratic) Kronecker
symbol, and the square root is chosen so that —IT/2 arg((cz +
d)1^2)
TT/2. Let
E(z, s) be the half-integral weight Eisenstein series
7eroo\r0(4)
Maass [47] showed in 1937 that the m t h Fourier coefficient of E(z, s) is essentially
equal to L(2s, Xm) where Xm is a quadratic character given by a Legendre symbol.
Here essentially equal means that this is correct up to Euler 2-factor, archi-
medean factors (suppressed from the notation) and most importantly correction
factors that adjust the formulas when m is not square-free. The correction factor
multiplying L(2s,x™) is a product of Dirichlet polynomials in
\v\~s
at the places
v such that oidv(m) 2.
Given any modular form, its Mellin transform is the Dirichlet series formed by
summing its Fourier coefficients. Siegel [55] applied a Mellin transform to E(z, s)
and observed that
/ [E(iy, s) const term j
ywdxy
«
Y^L(2s,Xri
Here the « is used to remind the reader that 2-factors, archimedian places and
correction factors are being suppressed. There is also an issue of normalizing the
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