SELBER6 TRACE FORMULA
17
where, for t t
C°(K+),
(1.12b) M^(s0) = J _ M^(S)[r+(s)S()) + rk(8,80)]^f.
Here r wRe(sQ+l), H is the usual Mellin transform on
R+,
and
2s"(3/2)r(s)r(s4)r(4s0+s4)r(-2s+2)
i.i3 rf(s,sft)= , , i—i^—i «
k
° r(s4sk- |)r(s- |sk-|)r(*
JB0-B+|)
(see [Z2],Proposition 3.3).
Suppose temporarily that (1.11) is valid for f e
CQ(R+);
we will return
to this question at the end of this section. Then let jL be the
characteristic function of the length interval [1,T] and jL CQ an
appropriate smoothing. ¥e substitute jL into the trace formula (1.11). As
usual, the complementary sreies are exponentially growing as T —* oo,while the
principal series terms are oscillatory. The latter may be estimated by the
remainder term in (1.8). The former contribute a finite sum of integrals
involving MJjjL . These may be evaluted very explicitly, thanks to (1.12b).
S
1)6
If we deform the line of integration leftwards, we pick up some residual terms
(1.14) Res
1
(E±r*(s,s.)WT (s).
s=2+s,
Combining (1.14) with the simple asymptotic
1 1 1
T(h*i),Jl
(1.15) ^(^j) - (^p e
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