SELBERG TRACE FORMULA 19

mentioned inthe introduction:

T

(1.18a) Mr(r,;t,T) = - ^/_T0p(r)E(.,£+ir), E(.,Jfir)dr

(if r is cuspidal)

T

(1.18b) = - ^/.

T

R[|E(.,^ir)|

2

, *+ir]dr

(if a = E(.,Jfir), e.g.)

Much ofthe analysis ofthis paper isdevoted toestimating the contribution

of these continuous spectral objects toVeyl laws and tothe error of the

prime geodesic theorem.

Finally, wewould like todiscuss a technical point raised above.

Namely, some ofour applications ofthe trace formula require the useof

non-compactly supported test functions. Most importantly, these test

functions occur when weuse the version (1.11) ofthe formula. Here j 6

CQ(R+),

but f = H" j t

CQ.

Ofcouse, it isquite commomplace touse test

functions ^ t CQ,butthe validity of (1.11) must beestablished for them.

The main point, aswill bediscussed indetail in§ 5,istojustify the trace

formulae for^ e OlR+) bya continuity argument. Namely, weview the trace

formula asan identity between linear functionals onCQ. Ve show that these

functionals arecontinuous ona certain Banach space inwhich CQ isdense.

The test functions ^ which are required inapplications will turn out to lie

in this space. This argument, implicit in [Z2], § 4,will bemade explicit in

§ 5.