ON THE DYNAMICAL SYSTEM GENERATED BY THE MÖBIUS TRANSFORMATION AT PRIME TIMES

The paper defines Th(N) as the unnormalized prime sum Th(N)=∑_{ℓ≤N, ℓ prime} e_p(h u_ℓ) and claims the power-decay bound max_{h∈F_p^*} |Th(N)| ≤ N^{-η} for N in [p^B, p^C] when the period t ≥ p^{3/4+ε} (Theorem 1.1) . This magnitude is dimensionally impossible: a standard second-moment identity over h shows max_h |Th(N)| ≥ π(N)·sqrt((p−t)/(p(p−1))). For the multiplicative map ψ(x)=g x with a primitive root g (hence t=p−1), this gives max_h |Th(N)| ≳ N/(p log N), contradicting the paper’s asserted upper bound for any fixed η>0 and any admissible choice of B,C>1. The body of Section 4 also pursues bounds of the schematic form Sp(M,N) ≪ N^{-ρ} for bilinear pieces (equation (4.3)), which mirrors the same normalization issue (the sums have about N terms) . The likely intended statement is either a normalized average π(N)^{-1}Th(N) ≪ N^{-η} or an unnormalized bound of size N^{1−η}/log N. Under such a corrected normalization, the paper’s Heath–Brown/Burgess machinery could plausibly deliver a power saving. As printed, however, Theorem 1.1 is false.