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