Counting functions over periodic orbits of a skew-product map

The paper defines weighted counting functions E_S(f,n), D_S(f,n), C_S(f,n) for the skew-product S, states Theorems 2.2–2.5 under the hypothesis E_S(f,n) ∼ λ_f^n, and proves them via Möbius inversion plus careful summations (lower/upper bounds for π_S(f,N), harmonic and Meissel-type sums, and a Dirichlet-series identity) . The candidate solution achieves the same conclusions more directly: via E = 1*D and D = μ*E one gets D(n) ∼ λ^n and hence C(n) ∼ λ^n/n, from which the three summation results follow, and the Dirichlet-series comparability is clarified to hold for truncated sums (resolving the divergence issue when λ>1). The only caveat is Theorem 2.5 in the paper: it is written with infinite Dirichlet series that diverge for λ_f>1 and uses 1/ζ(z) instead of the exact identity’s 1/ζ(z+1); both are harmless under comparability once one states the result for truncated sums, as the model does. Thus the paper’s claims are essentially correct but would benefit from tightening the statement of Theorem 2.5; the model’s proof is correct and somewhat cleaner on this point.