Zalcman's Bessel problem: does $J_0(z)=1$ have at most one solution on each ray? (Problem 2.45)
Statement
Let $J_0$ be the Bessel function of the first kind of order zero, $J_0(z)=\sum_{m=0}^{\infty}\frac{(-1)^m}{(m!)^2}\bigl(\tfrac{z}{2}\bigr)^{2m}$. Is it true that the equation $J_0(z)=1$ has at most one solution on each ray $\{re^{i\theta}:r>0\}$ from the origin? Equivalently, for every fixed $\theta$, does the entire function $r\mapsto J_0(re^{i\theta})-1$ have at most one zero with $r>0$? Classical asymptotics already show that each ray carries at most finitely many solutions; the open question is whether 'finitely many' can be sharpened to 'at most one'.
Acceptance. FULLY RESOLVES: EITHER a proof that for every $\theta$ the ray carries at most one solution of $J_0(z)=1$ (thereby voiding the Delsarte-Lions exceptional set), OR an explicit ray $\theta$ with two certified distinct solutions $r_1\ne r_2>0$ of $J_0(re^{i\theta})=1$ (a finite, interval-arithmetic-checkable counterexample). ADVANCES: a rigorous reduction of the count to 'at most two' on every ray; a certified determination of the extremal ray (where two solutions nearly collide); or a proof valid for a sector of $\theta$-values. Floating-point-only evidence does NOT qualify.
Background
Posed by L. Zalcman; Problem 2.45 in W. K. Hayman & E. F. Lingham, Research Problems in Function Theory (Fiftieth Anniversary Edition, Springer 2019), source of record arXiv:1809.07200, whose Update 2.45 states 'No progress on this problem has been reported to us.' An affirmative answer would show that the exceptional set in a theorem of Delsarte and Lions (a two-radius / mean-value theorem) is void. The only known bound is the asymptotic 'finitely many per ray', from $J_0(z)\sim\sqrt{2/(\pi z)}\cos(z-\pi/4)$; no published partial result narrows this to 'at most one' on any ray. Certified interval arithmetic can rigorously count the zeros of $J_0(re^{i\theta})-1$ on bounded $r$-segments, while the uniform asymptotic tail handles $r\to\infty$; the decisive step is a short argument-principle / monotonicity argument in the transition region, aided by a numerical survey of $\theta$ to locate the extremal ray where two solutions nearly collide. (A. Eremenko's 2021 Bessel notes treat zero-asymptotics of $J_\nu$, not this uniqueness question.) Vetted open as of 2026-07-06 (high confidence; Zalcman d. 2022, no posthumous resolution found).
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Hayman & Lingham, Research Problems in Function Theory (New Edition) - Problem 2.45 (Zalcman) | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.