DIRICHLET DYNAMICAL ZETA FUNCTION FOR BILLIARD FLOW

The paper proves a clean non-entireness criterion (Theorem 1.1) for the Dirichlet dynamical zeta function η_D(s) under two hypotheses: exponentially small one-sided gaps λ_{m_j}-λ_{m_j-1} ≥ C e^{-δ λ_{m_j}} with δ>h+1 and exponentially small tails |∑_{n≥m_j} a_n| ≥ e^{γ λ_{m_j}} with γ<σ_c<0. The proof assumes, for contradiction, that η_D is entire, derives uniform subpolynomial growth on vertical strips via a local trace formula and Phragmén–Lindelöf, then invokes Hardy–Riesz ‘typical means’ and Kuniyeda’s abscissa formula to get super-exponential decay of R(u)=∑_{λ_n>u} a_n(λ_n-u). This contradicts the gap–tail lower bound, completing the argument. The steps, including the key bounds (2.3)–(2.4), Proposition 2.1 (σ_k=−∞), and the Kuniyeda-based contradiction, are explicit and coherent in the note (see Theorem 1.1 and its proof, as well as the local trace formula discussion) . By contrast, the model’s outline sets up a Laplace–Stieltjes framework and correctly identifies that a naive L1 argument is too weak given N(x)~e^{hx}/(hx) , but it never supplies the decisive nonextendability lemma (Fabry/Mandelbrojt type) it relies on. The kernel localization and “difference quotient blows up” claims are asserted without rigorous control of the remainder or a cited theorem that would close the argument. Hence the model solution is incomplete, whereas the paper’s proof is complete and correct.