Abstract

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

Secondary 58G

(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.

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