Ve use harmonic analysis on a finite area hyperbolic surface T\h to
optimally sharpen the Bowen equidistribution theory of closed geodesies. This
theory is concerned with the series f (r,T) = E fr5 where a is an
automorphic form on r\PSL2(R), and where the sum runs over all closed
geodesies 7 of length L less than T. Using a generalization of the Selberg
trace formula, we obtain an asymptotic expansion for each f(r,T), which
reverts to the standard prime geodesic theorem if a = 1. The principal term
vanishes if a is orthogonal to 1, giving a new proof that closed geodesies are
uniformly distributed with respect to Haar measure, together with an
exponential rate of equidistribution.
One of the main steps in deriving these asymptotic expansions is to
estimate certain dual spectral sums (M+N)(r,T), which revert to the usual
winding number of the scattering phase plus eigenvalue counting function if
a = 1. When a is an Eisenstein series and Y = PSL2(ff), our estimate gives a
weak (signed and averaged) version of the Lindelof hypothesis for
Rankin-Selberg zeta functions.
This paper is a continuation of our earlier work on compact hyperbolic
surfaces (cf. [Z.2]).
(1) Subject Classification: Primary 11F, 58F
(2) Key words: finite area hyperbolic surface, closed geodesic, cusp form,
Eisenstein series, Rankin-Selberg zeta function.
Received by editor March 27, 1990 and in revised form March 4, 1991.