The Equation J_0(z) = 1 Has at Most One Solution on Each Ray: An Answer to a Question of Zalcman

L. Zalcman asked whether the equation J_0(z) = 1, where J_0 is the Bessel function of order zero, has at most one solution on each ray from the origin. The question is Problem 2.45 in Hayman and Lingham's Research Problems in Function Theory, where it is noted that an affirmative answer would show that the exceptional set in a two-circle theorem of Delsarte and Lions is void. We prove that the answer is yes. Apart from the double solution z = 0, the solutions are ±z_m and ±z̄_m (m ≥ 1), all simple, where z_m lies in the open first quadrant and π/2 > arg z_1 > arg z_2 > … > 0. Hence every open ray from 0 contains at most one solution. (On closed rays the statement fails trivially, because 0 is a solution.) For Re z ≥ 60 the proof uses Olver's explicit error bounds for Hankel's expansion, a contraction mapping, Rouché's theorem and a differential inequality for the arguments of the approximate solutions. For the rectangle [0, 60] × [0, 5] it uses a certified computation in exact integer ball arithmetic (an argument-principle count, Rouché discs and exact comparisons of arguments), which was reproduced by two independent interval-arithmetic computations. Consequently, the set of positive quotients of nonzero solutions of J_0(z) = 1 is {1}, and Delsarte's two-circle theorem in the plane holds for every pair of distinct radii: a continuous function on ℝ² with the circle mean-value property for two distinct radii is harmonic. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-022-2045 (W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, Fiftieth Anniversary Edition, Springer 2019, Problem 2.45, attributed to L. Zalcman).