SELBER6 TRACE FORMULA
where, for t t
(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
i.i3 rf(s,sft)= , , i—i^—i « —
° r(s4sk- |)r(s- |sk-|)r(*
(see [Z2],Proposition 3.3).
Suppose temporarily that (1.11) is valid for f e
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).
If we deform the line of integration leftwards, we pick up some residual terms
Combining (1.14) with the simple asymptotic
1 1 1
(1.15) ^(^j) - (^p e