16 STEVEN ZELDITCH

(1.10) |Nr(r,T)| « Nr(T)

(s,m)

is all that one needs. The generalization of (1-7) and (1.8) to the finite

area case will be the object of § 4; perhaps surprisingly, even the extension

of (1.10) is non-trivial.

Generalization of the prime geodesic theorems (1.8) to the finite area

case is the primary goal of this paper. Much of what occurs in the compact

case carries over verbatim to the general finite area case. This applies in

particular to the hyperbolic and discrete spectral terms of the trace formula.

Note that (1.8) is, up to the error term, a relation between the hyperbolic

terms and the complementary series spectral terms. Since the latter always

occur discretely, one expects (1.8) to be valid (up to the error term) in the

finite area case. This indeed proves to be the case.

The analysis of the hyperbolic and discrete spectral terms in [Z2] will

be relied on heavily in this paper. For the sake of completeness and

intelligibility, let us summarize here the main steps leading to (1.8).

The first step is to invert the non-standard HC transforms arising in the

hyperbolic terms of the trace formula (1.1). Setting ^ = Hffl 6, where / e CQ,

the trace formula in the compact case becomes:

(1.11) E(0p(,)u u )Smo Hm^(Sj) = S ( I ^ ( L

?

).

J

t"hyp

U

For concreteness, let us assume m = 0 (the other cases are similar). The

transform S o H" can then be given the quite explicit expression:

def

c

(1.12a) S o

H~1s

\ = IT

v ' m m,

s,

'