142 J.W. COGDELL, £-FUNCTIONS FOR GLn
work with the incomplete £-function Ls(s,1f x 7r^1 ) where Sis as at the beginning
of this Lecture. Then we can write
LS (s , 7f x 7r^1 ) = II L(s, 1fv X 7r~) = II det(J - q_;;-^5 A11'v ® A'Jl'J-^1
v~S v~Swith absolute convergence for Re(s) >> 0.
Recall that an infinite product Il(l +an) is absolutely convergent iff the asso-
ciated series L log(l +an) is absolutely convergent.
Let us write
We have seen that lμv,i I < q~/2 and lμ~,j I < q~^12. Then
log L(s, 1fv x 7r~) = -L log(l - μv,iμ~,jq,;;-^8 )
i,j
= "'~ (μv,iμ~)d = ~ tr(A~J tr(A~)
L., L., dqds L., dqds
i,j d=l v d=l v
with the sum absolutely convergent for Re( s) > > 0. Then, still for Re( s) > > 0,(^1) og (Ls( s, 7f x 7f ')) -_ L., "'~ L., tr(A~J dqdtr( s A~).
v~S d=l v
If we apply this to 1f^1 = 7F = 'if we find
oo I tr( Ad )12
log( Ls (s, 7r x 7F)) =LL d ;;
v~S d=l qv
which is a Dirichlet series with non-negative coefficients. By Landau's Lemma
this will be absolutely convergent up to the its first pole, which we know is at
s = 1. Hence this series, and the associated Euler product L(s, 7r x 'if), is absolutely
convergent for Re( s) > 1.
An application of the Cauchy- Schwatrz inequality then implies that the series
(^1) og (Ls( s' 7f X 7f ')) = L., "'~ L., tr(A~J dqds tr( A~)
v~S d=l v
is also absolutely convergent for Re(s) > 1. Thus L(s, 7r x 7r^1 ) is absolutely conver-
gent and hence non-vanishing for Re(s) > 1.
To obtain the non-vanishing on the line Re(s) = 1 requires the technique of
analyzing £-functions via their occurrence in the constant terms and Fourier coef-
ficients of Eisenstein series, which we have not discussed. They can be found in the
references [44] and [75, 76] mentioned above.
4.5. The G eneralized Ramanujan Conjecture (GRC)
The current version of the GRC is a conjecture about the structure of cuspidal
representations.