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In this note we give an alternative proof of Sharp's dynamical analogue of Merten's theorem for hyperbolic flows. Our use of a tauberian theorem of Agmon also allows us to get better error terms. 0. INTRODUCTION In an earlier paper, Parry and the author proved that the asymptotic distribution of closed orbits for hyperbolic flows is analogous to the prime number theorem in number theory [4]. More recently, Sharp has derived the corresponding analogues for closed orbits of Merten's theorems on prime numbers [5]. (Essentially, these results on the distribution of prime numbers are weaker but more accessible than the prime number theorem.) The purpose of this short note is to present an alternative derivation of Sharp's results which has the significant advantage that we are able to obtain far better error estimates. Let qt: A -A be a (weak-mixing) hyperbolic flow restricted to a basic set A, with topological entropy h > 0. Let T denote a closed orbit of least period i(T) and we shall denote N(T) = eh4(T).

Agmon’s complex Tauberian theorem and closed orbits for hyperbolic and geodesic flows