difficult. In [Z.5] we show at least that the sum in (0.13) is o(r). Our
method could give also the slightly better result 0(T /InT).
The Veyl laws (0.11) and (0.12) are crucial in the proof of (0.4). Since
(0.12) is of higher order than (0.11), it should not be surprising that the
remaninder in (0.4) is of higher order than in (0.2). This large remaninder
is also due our crude estimate in the growth of the periods E(-,s) as
L —» OD (Res = 1/2). Based on the fact that a closed geodesic can only go a
distance of e ' away from a compact part of T\h into an end, we bound
f ? 7
E(-,s) by 0(e ') It is quite likely that a typical closed geodesic only
goes a distance L into an end (cf. [Su]); so this aspect of the remainder
estimate could probably be improved a great deal.
As mentioned above, this paper is an extension of our earlier work on a
compact hyperbolic surface [Z.2]. The results of that work will be summarized
in §1, but will otherwise be assumed without proof. Thus, this paper is far
from self-contained.
Although we only consider hyperbolic surfaces in [Z.2] and in this paper,
it seems reasonable that similar asymptotic expansions should exist on a
general cofinite quotient of a rank one symmetric space. In fact, it seems
likely that they should occur on any locally symmetric quotient for which a
prime geodesic theorem could be proved. In particular, it seems reasonable to
predict that if a prime geodesic theorem could be proved for compact regular
flats in higher rank quotients, then these flats could be shown by our more
general trace formulae to be equidistributed. However, the analysis would
have to be very long and clumsy, since it would involve a case-by-case study
of automorphic forms.
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