Finite positivity of Li-type coefficients: extremal configurations in the Möbius lens

Let ξ be an entire function of order less than 2 satisfying a functional equation ξ(1−s) = w ξ*(s) with |w| = 1, with all zeros in the strip 0 ≤ Re ρ ≤ 1 and with a zero-counting estimate of the usual shape. Brown (2005) gave two finite-index results relating the positivity of the first Li-type coefficients Re b_n of ξ to zero-free regions; the printed proofs of both rest on a lemma (his Lemma 5) whose proof later readers have found defective; nothing here depends on either theorem being established, since what is analysed is the shape of the proof of the second. Transporting the zeros by the Möbius map z(ρ) = ρ/(ρ−1) turns the relevant zero-free region into a disc and the coefficients into a sum of per-zero terms 1 − cosh(ku) cos(kθ), a known form (Brown; Dehghani; Pavesi) that we do not claim.\n\nOur main result is an admissible family of off-line zero orbits on which the per-zero bound 2, which an upper-bound argument at the T² scale would naturally assume, fails by a factor independent of T; we also determine the limit of a lower bound for the associated functional along a one-parameter family. Our second result is a no-go for the power-sum step of Brown's argument: when, as there, the power-sum theorem is applied with the exponents j·k₀, the range factor cannot be shortened multiplicatively below the number of points, so that no bound of the form O(T² log^α T) is reachable by that route; the proof uses only the extremal power-sum identity of Beukers and Tijdeman together with the strip hypothesis. Alongside these we record two facts about the coordinate: the Beukers–Tijdeman family scaled to the boundary circle is admissible there for every integer q with 3 ≤ q ≤ ⌊2π/arctan(1/T)⌋ and for no larger q; and at a suitable common modulus in (1, r(T)], depending on the angles, the strip hypothesis and the radial condition impose no restriction on angles ≠ 0, so that an additive shortening resting on those two hypotheses alone would have to be proved for arbitrary unimodular configurations avoiding the point 1.\n\nThe framework uses only the functional equation, the strip and the counting estimate, with no Euler product and no arithmetic input; its class contains functions with zeros off the critical line. Nothing in this paper bears on the Riemann hypothesis.