# Verification report — AMR-022-2045 (Hayman–Lingham Problem 2.45, L. Zalcman: does J_0(z) = 1 have at most one solution on each ray?)

Verification date: 2026-10-03.

**Verdict.** The answer is yes. Apart from the double solution z = 0, the solutions of J_0(z) = 1 are ±z_m and
±conj(z_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. The set H_2 of positive quotients of nonzero solutions
is {1}, so Delsarte's two-circle theorem in the plane holds for every pair of distinct radii. The proof is
computer-assisted on the rectangle [0,60] × [0,5]; that computation was reproduced by two independent AI-assisted
verification runs with their own interval-arithmetic programs, the second with a different counting method. The
note is unrefereed.

## Statement checked
- **Primary source.** W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, Fiftieth Anniversary
  Edition, Problem Books in Mathematics, Springer, Cham, 2019, doi:10.1007/978-3-030-25165-9; arXiv:1809.07200
  (the arXiv TeX source was read, twice, by independent runs).
  - Problem 2.45, attributed to L. Zalcman, asks whether J_0(z) = 1 "has at most one solution on each ray from
    the origin".
  - The problem text adds that an affirmative answer would make the exceptional set in a theorem of Delsarte and
    Lions (Comment. Math. Helv. 33 (1959) 59–69, doi:10.1007/BF02565907) void, and that asymptotic estimates
    allow at most finitely many solutions on each ray.
  - Update 2.45 in the arXiv version (2018) states that no progress on the problem had been reported to the
    authors. The printed 2019 book was not seen.
  - By the tables of the arXiv version, the problem first appeared in the list "New problems" of the Proceedings
    of the Symposium on Complex Analysis, Canterbury 1973 (LMS Lecture Note Series 12, 1974). That list was not
    seen.
- **Corpus record.** ulamai/UnsolvedMath version 1.6.0, AMR-022-2045, status `open`. Its statement agrees with
  the source.

## Readings
| Reading | Answered? | Witness |
|---|---|---|
| open rays {t e^{iθ} : t > 0}, i.e. nonzero solutions (intended reading) | yes, affirmative | Theorem 1.1 |
| closed rays, including the origin | trivially negative | z = 0 is a (double) solution and lies on every closed ray; every ray through some z_m contains two solutions |
| the exceptional set of the two-circle theorem: is H_2 = {1}? | yes | Corollary 1.3 |
| the planar two-circle theorem for every pair of distinct radii | yes, via Delsarte's theorem (Theorem 1.2, quoted) | Corollary 1.4 (for r_1 = r_2 it fails: e^{i z_1 x_1 / r}) |
| sphere means in R^n, n ≠ 2 (Λ_n(z) = 1, e.g. sin z = z for n = 3) | not treated | Remark 6.2 |

## Results in the paper
- **Lemmas 2.1–2.3.** Symmetries; g = J_0 − 1 is negative on the real axis and positive on the imaginary axis
  (except at 0); g = −(z²/4)S(z²/4) with |S − 1| < 0.0643 on |z| ≤ 1; |J_n(z)| ≤ I_0(|Im z|) for n = 0, 1, 2.
- **Lemma 2.4.** Hankel's expansion with Olver's error bounds, used in the form of DLMF §10.17(iv)
  (10.17.13–10.17.15; the page reference Olver 1997, pp. 266–267, is taken from DLMF; the book was not seen),
  for ν = 0, ℓ = 1 on the closed first quadrant: |R^+| ≤ (1/(4r)) e^{1/(4r)} and |R^−| ≤ (π/(8r)) e^{π/(8r)}.
  - The paths of variation must have Im t monotone (the Volterra kernel, (1 − e^{∓2i(z−t)}) times a constant of
    modulus 1/2, is then bounded by 1). The DLMF text prints |Im t|, which cannot hold literally for V_{z,−i∞}
    when Im z > 0, nor for DLMF's own bound for V_{z,i∞} when −π/2 ≤ arg z < 0.
  - Explicit admissible paths give V_+ ≤ 1/r (ray, arc, imaginary axis) and V_− ≤ θ/(r sin θ) ≤ π/(2r)
    (horizontal to the right, then down).
- **Section 3 (Re z ≥ 60).**
  - Lemma 3.1: zero-free zones |u| ≥ 2 (r ≥ 5), |u| ≤ 1/2 (r ≥ 60), and the strip 0 ≤ Re z ≤ 60, Im z ≥ 5.
  - Lemma 3.2: T_t(z) = 2πt + π/4 + (i/2) Log(2πz) is a 1/120-contraction of Ω = {Re z ≥ 60, Im z ≥ 0}, with
    fixed point F(t), F' = 2πF/(F − i/2).
  - Lemma 3.3: every zero in Ω satisfies |z − F_m| < 0.576/(2πm) for some m ≥ 10.
  - Proposition 3.4: exactly one simple zero z_m in each D_m = D(F_m, 1/(2πm)), and no others in Ω.
  - Proposition 3.5: m²(arg F_m − arg F_{m+1}) ≥ 0.3196 > 0.0507 ≥ m² × (angular widths of D_m and D_{m+1}).
  - Lemma 3.6: Im F_10 < 3.0193 and sup arg over D_10 ≤ arg b* < 0.04829.
- **Proposition 4.1 (computer-assisted).** Exactly nine zeros in [0,60] × [0,5] minus {0}, simple, in discs of
  radius 2^-150, with decreasing arguments; arg z_9 = 0.05127... > arg b*.
- **Theorem 1.1, Corollaries 1.3 and 1.4** follow (Section 5). Theorem 1.2 is Delsarte's two-circle theorem,
  quoted in the form of the zbMATH summary of Berenstein–Gay (1986): it follows from the local theorem stated
  there, applied on discs of radius R > r_1 + r_2. The summary prints the function defining H_n with an unclosed
  parenthesis and without "− 1"; it is read as Λ_n − 1, the only reading compatible with the finiteness of H_n
  stated there, with the examples e^{iλx_1} and with the remark in the problem book. The exponential examples
  show only that the hypothesis of Theorem 1.2 is necessary.

## Computations (scripts and outputs in reproducibility/)
- **Finder** (`finder/`).
  - `cert_bounded_2045.py` with `ball_2045.py`: exact integer ball arithmetic (radius unit 2^-320). The right and
    top sides of the rectangle are walked in 31 + 256 certified steps; the argument changes by 19π, so there are
    9 zeros. Nine Rouché discs of radius 2^-150 are certified (|g(ẑ)| ≤ 2.2e-73, |g'(ẑ)| ≥ 0.94), and the
    arguments are ordered by exact cross products (≥ 9.64). The junction with the tail is checked
    (Im(ẑ_9 conj b*) = 10.78 > 0). About 1 s.
  - `tail_constants_2045.py`: [T1]–[T7] and [Q] in exact rationals; ALL CHECKS PASSED.
  - `sanity_scipy_2045.py` (not part of the proof): zeros for m ≤ 20000; a floating-point argument principle
    (9.000000); Olver ratios ≤ 0.50 and 0.34; actual Rouché margins 0.99 and 0.20 against the proof bounds
    0.9837 and 0.5641.
- **First independent verification run** (`independent_run/`, AI-assisted, own code, mpmath interval arithmetic
  at 420 bits).
  - Full-rectangle argument principle on [−X, X] × [−5, 5]: 38 = 2 + 4·9 zeros for X = 60 and
    50 = 2 + 4·12 for X = 80.
  - Rouché discs of radius 2^-100 for z_1..z_12, with argument gaps from 0.0997 down to 0.0030.
  - (2πm)|z_m − F_m| ≤ 0.1995, 0.1998, 0.2000 for m = 10, 11, 12.
  - Junction: arg z_9 > 0.05127 > 0.04733 ≥ sup arg over D_10.
  - All tail inequalities re-checked, plus direct checks for 450 values of m up to 1e10.
- **Second independent verification run** (`independent_run_2/`, AI-assisted, own code written from the paper
  only, mpmath interval arithmetic at 240 bits).
  - Count without the argument principle, symmetry or the axis lemmas: the closed rectangle B is covered by
    [0,1]² (where h = (J_0 − 1)/z² ≠ 0), nine Rouché discs of radius 1/8 (one simple zero each) and 969 squares
    with |g(c)| − |g'(c)|s − s² I_0''(η)/2 > 0 (no zero). Exactly nine zeros in B \ {0}, none on ∂B. About 11 s.
  - Rouché discs of radius 2^-110 for z_1..z_12; every digit of Table 1 confirmed; arg z_1 > … > arg z_12.
  - F_10, F_11, F_12 enclosed; (2πm)|z_m − F_m| ≤ 0.199491, 0.199753, 0.199972; z_m ∈ D_m.
  - Junction: sup arg over closed D_10 ≤ 0.0473228 < 0.0512714 ≤ arg z_9; arg b* = 0.0482830 < 0.04829.
  - [Q], [T1]–[T7] and auxiliary inequalities re-checked (38 checks); direct checks of Lemma 3.3 and
    Propositions 3.4–3.5 for 3038 values of m in [10, 1e15]; arg F_m − arg F_{m+1} ≥ G(m) for m = 10..59.
  - Not part of the proof: Olver ratios on a polar grid including both edges (max 0.4997 and 0.3383); path
    variations of Lemma 2.4 by quadrature; Remark 6.1 at sample m up to 20001.
- **Re-runs.** All scripts were re-run from the packaged copies when the package was assembled, and again from an
  extracted copy of the final source archive. The outputs agree with the originally recorded ones apart from
  timing lines.

## Independent verification runs
Both runs were AI-assisted verification runs, not human peer review.

**First run** (2026-10-03). It re-derived the analytic inequalities, re-ran the finder's scripts, and wrote its own
interval-arithmetic code with a different contour, step criterion and argument accounting. Verdicts: statement
fidelity, proofs, computations and answer as posed CONFIRMED; novelty CONFIRMED as far as can be checked (no
priority claim); presentation CONFIRMED_WITH_FIXES. The fixes it required were applied: Olver's path condition and
admissible paths (Lemma 2.4); the precise form of Delsarte's theorem with H_2 (Theorem 1.2) and the remark that
the exponential examples show only necessity; publication of the code; the 2023 "Zalcman problem" papers;
cosmetic fixes (step bound π/6, a stale section reference, internal remarks).

**Second run** (2026-10-03). It re-read the problem in the arXiv source and DLMF §10.17, re-derived every proof,
re-ran all programs from the source archive, and wrote its own programs (`independent_run_2/`).

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (wording of Update 2.45 corrected) |
| Proofs | CONFIRMED (no mathematical error; one clause added to the proof of Theorem 1.1) |
| Computations | CONFIRMED (count reproduced by a different method; ordering and junction certified) |
| Answer as posed | CONFIRMED (affirmative for the intended open-ray reading) |
| Novelty | CONFIRMED as far as can be checked (no priority claim) |
| Presentation and sourcing | CONFIRMED_WITH_FIXES (all applied) |

Fixes required by the second run, all applied:
1. Record the second run (abstract, Method paragraph, Section 3 preamble, Section 4.1, Verification paragraph, this
   report, `reproducibility/independent_run_2/`).
2. House style: the independent checks are described only as AI-assisted verification runs.
3. Update 2.45: "no progress had been reported to the authors" (arXiv version, 2018), not "no progress had been
   made"; the problem's two remarks attributed unambiguously to the problem text.
4. Theorem 1.2: state that it is quoted from the zbMATH summary of Berenstein–Gay (garbled formula read as
   Λ_n − 1), that it follows from the local theorem stated there, and that Delsarte (1958) and Delsarte–Lions
   (1959) were not seen; the reviews of Volchkov (1995, ball means) and (1996) are no longer cited as sources of
   the formulation.
5. The content of Volchkov–Volchkov, Anal. Math. Phys. 13 (2023) is no longer asserted (its abstract was not
   accessible); only its zbMATH record is described.
6. Olver's bound is stated to be used in the form of DLMF §10.17(iv); the book itself was not seen.
7. Proof of Theorem 1.1: a solution F with Re F ≥ 60 has Im F = ½ ln(2π|F|) > 0, so lies in Ω, where it is
   unique.
8. Abstract: the problem book is credited to Hayman and Lingham.

## Relation to the literature, novelty and scope
- **Searches (October 2026).** arXiv, Crossref, OpenAlex, zbMATH, Semantic Scholar and the web (one web search per
  run). Queries covered the equation J_0(z) = 1, Bessel a-points, the two-circle (two-radius) theorem and its
  exceptional set, Delsarte, Zalcman, and Volchkov's two-radii papers. The 27 works citing Delsarte–Lions 1959
  listed by OpenAlex were screened.
  - No work claims H_2 = {1} or settles the rays of J_0(z) = 1.
  - The arXiv version of the problem book (2018) reports no progress.
  - The zbMATH summary of Berenstein–Gay 1986 (Israel J. Math. 55, 267–288) describes H_n as finite and
    containing 1.
  - Volchkova–Volchkov, Ural Math. J. 9 (2023) 187–200: by its zbMATH summary and journal page, it treats the
    reconstruction of a distribution f from f∗σ_{r_1} and f∗σ_{r_2} (a different problem of Zalcman).
  - Volchkov–Volchkov, Anal. Math. Phys. 13 (2023), Paper 72: abstract not accessible; its zbMATH record
    (classification and reference list) does not show which problem of Zalcman is meant, and the reference list
    does not include the problem books.
  - Other works found with "problem of Zalcman" in the title concern different problems (a Morera-type problem,
    Volchkov 1993; Hayman–Lingham Problem 7.29, arXiv:2608.02546).
- **Caveats.** The following were not accessed: Delsarte (C. R. Acad. Sci. Paris 246 (1958)); Delsarte–Lions
  (1959); Berenstein–Gay (1986; zbMATH summary only); the Volchkov papers of 1995 and 1996 (zbMATH reviews only);
  Zalcman's survey (Amer. Math. Monthly 87 (1980)); the Netuka–Veselý survey (1994); Flatto (1965); Zalcman
  (1972); Olver's book (1997); the 1974 Canterbury problem list; the full text of the 2023 Volchkov papers. The
  statement of Delsarte's theorem follows the zbMATH summary of Berenstein–Gay. This negative search is not a
  proof of priority. The method (explicit asymptotics plus a finite certified computation) is standard, and the
  result may be within reach of experts.
- **Scope.** The note settles Problem 2.45 for the intended open-ray reading and proves H_2 = {1}. The analogous
  question for sphere means in R^n, n ≠ 2, is not treated.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/