Estimates on the number of orbits of the Dyck shift.

In this paper, we get crucial estimates of fundamental sums that involve the number of closed orbits of the Dyck shift. These estimates are given as the prime orbit theorem, Mertens' orbit theorem, Meissel's orbit theorem and Dirichlet series. Different and more direct methods are used in the proofs without any complicated theoretical discussions. [ABSTRACT FROM AUTHOR]