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Bessel functions and Weyl's law for balls and spherical shells

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Pith's one-line read The paper establishes that the Weyl remainder for balls and spherical shells in every dimension $d \ge 2$ is bounded by $\mu^{d-2+2\theta_*+\varepsilon}$, with $2\theta_* \approx 0.628966$, in both Dirichlet and Neumann cases.

desk verdict Strong technical paper whose improved Weyl exponent rests on a sketched estimate outsourced to an unpublished preprint by the last author; referee before trusting the new bound. read the letter →

arxiv 2412.14059 v1 pith:STU32ZNQ submitted 2024-12-18 math.CA math.NTmath.SP

Jingwei Guo , Tao Jiang , Zuoqin Wang , Xuerui Yang This is my paper · ORCID
classification math.CAmath.NTmath.SP MSC 35P2042B2011P2133C10
keywords WeyllaweigenvaluecountingBesselfunctionsultrasphericalsphericalshellsGausscircleproblemweightedlatticepointdecouplingestimates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves new upper bounds for the remainder in Weyl's law for the Dirichlet and Neumann Laplacians on Euclidean balls and spherical shells in all dimensions $d \ge 2$. For balls the remainder is shown to be $O_\varepsilon(\mu^{d-2+2\theta_*+\varepsilon})$, and for spherical shells it is $O(\mu^{d-2+2/3})$ in general, improving to $O_\varepsilon(\mu^{d-2+2\theta_*+\varepsilon})$ when $\pi^{-1}\arccos(r/R)$ is rational. The exponent $2\theta_* = 0.628966\ldots$ improves the older $131/208 = 0.629807\ldots$. To reach these bounds, the paper develops uniform asymptotics and zero-counting results for cross-products of Bessel functions and of derivatives of ultraspherical Bessel functions, results that are presented as independently useful. If the central technical estimate holds, the shell results are new in all dimensions and the ball results cover the Neumann case for the first time.

What carries the argument

The load-bearing object is the planar domain $\Omega$ bounded by the graph of $G(x)=Rg(x/R)-rg(x/r)$ and its Minkowski functional $F$; the positive zeros of the Bessel cross-products $f_\nu$, $g_\nu$, and $h_{\nu,\delta}$ are approximated as $F(\nu,k+\tau_{\nu,k})$ with uniform remainder terms. That approximation turns eigenvalue counting into weighted lattice-point counting in $\mu\Omega$, and the exponent $2\theta_*$ enters through Theorem 6.1, an estimate for sums of rounding errors with the sawtooth function that extends the recent Gauss-circle bound to the functions arising in this spectral problem.

What would settle it

Take $F(x)=\sqrt{x+1}$ on $[1/2,2]$; it satisfies $|F^{(j)}(x)| \asymp 1$ for $j=1,2,3$ while $F'F'''-3(F'')^2 \equiv 0$, so the nondegeneracy condition of the cited preprint fails exactly in the way Theorem 6.1 claims is harmless. For $M = T^{141/328+c}$ with a small $c>0$, evaluate the rounding-error sum in (6.3) for large dyadic $T$; if any value exceeds $T^{\theta_*+\varepsilon}$, Theorem 6.1 is false and the improved Weyl bounds do not follow.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for $d \ge 2$ and any $\varepsilon > 0$, the Weyl remainder for a ball satisfies $R_B(\mu) = O_\varepsilon(\mu^{d-2+2\theta_*+\varepsilon})$, and for a spherical shell satisfies $R_S(\mu) = O(\mu^{d-2+2/3})$, improved to $O_\varepsilon(\mu^{d-2+2\theta_*+\varepsilon})$ when $\pi^{-1}\arccos(r/R) \in \mathbb{Q}$, in both Dirichlet and Neumann cases. The proof converts eigenvalue counting into weighted planar lattice-point counting in a domain built from the function $G(x)=Rg(x/R)-rg(x/r)$, with $g(x)=(\sqrt{1-x^2}-x\arccos x)/\pi$, and then applies a recent exponential-sum estimate from the Gauss circle problem, extended in Theorem 6.1 to the functions that arise here. The paper also establishes that all zeros of the relevant cross-product of derivatives of ultraspherical Bessel functions are real and simple, and counts those zeros inside a large circle.

Load-bearing premise

The improved exponent $2\theta_*$ stands on Theorem 6.1, a sketched extension of an exponential-sum estimate from the preprint [27] that removes a nondegeneracy condition in the range $T^{141/328+c} \le M \le T^{1/2}$; if that extension fails, only the weaker $O(\mu^{d-2+2/3})$ shell bound and the older $131/208$-type bounds remain secure.

Editorial extensions

If this is right

  • For every $d \ge 2$, the Dirichlet and Neumann eigenvalue-counting remainders for balls improve to $O_\varepsilon(\mu^{d-2+2\theta_*+\varepsilon})$, beating the previously known $131/208$-based bound in all dimensions.
  • For spherical shells, the general remainder $O(\mu^{d-2+2/3})$ holds for both boundary conditions, and under the rationality condition $\pi^{-1}\arccos(r/R) \in \mathbb{Q}$ it improves to $O_\varepsilon(\mu^{d-2+2\theta_*+\varepsilon})$, a result that is new in dimensions $d \ge 3$.
  • The uniform zero approximations, such as $x''_{\nu,k} = F(\nu,k+\tilde\tau_{\nu,k}) + O((\nu+k)^{-1})$ in appropriate ranges, provide explicit approximations of eigenvalues with error terms independent of the quantum numbers $\nu$ and $k$.
  • The proof that the cross-product of derivatives of ultraspherical Bessel functions has only real and simple zeros, with an exact count inside large circles, supplies a spectral ingredient that can be reused in other boundary-value problems with spherical symmetry.
  • The reduction to weighted planar lattice-point counting, with weights coming from eigenvalue multiplicities, gives a template for transferring further improvements of the Gauss circle problem exponent to Weyl remainders.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: The same weighted lattice-point reduction should carry the $2\theta_*$ exponent to Robin boundary conditions on balls and spherical shells, since the boundary-condition parameter enters only through the Bessel cross-product whose zeros are already treated with uniform estimates.
  • Editorial inference: The uniform zero approximations developed here could be used to certify numerical eigenvalue computations for spherical shells with explicit error bounds, a by-product the paper does not pursue.
  • Editorial inference: If Theorem 6.1 is fully verified, any future improvement of the Gauss circle or Dirichlet divisor exponent would automatically improve these Weyl remainders through the same reduction, making the spectral problem a direct consumer of exponential-sum progress.
  • Editorial inference: The reverse inequality noted in Remark 4.3 suggests that sharper Weyl bounds on shells could in turn yield new lattice-point estimates for the planar domain $\Omega$, reversing the usual direction of transfer.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper develops uniform asymptotics and zero-counting results for cross-products of Bessel functions and ultraspherical Bessel functions, and applies them to the two-term Weyl law for Euclidean balls and spherical shells in R^d. Theorem 1.1 claims a remainder O_epsilon(mu^{d-2+2theta_*+epsilon}) for balls in all dimensions and both Dirichlet and Neumann boundary conditions, with 2theta_* approximately 0.628966, improving the previous 131/208 approximately 0.629807; for shells it gives O(mu^{d-2+2/3}) in general and the same improved exponent when pi^{-1} arccos(r/R) is rational. The proof reduces eigenvalue counting to weighted planar lattice-point counting via the Bessel zero asymptotics, and then applies a new exponential-sum estimate (Theorem 6.1).

Significance. If the proof of Theorem 6.1 is completed, the paper's results are significant: they improve the best known upper bounds for the Weyl remainder on balls in all dimensions, extend the planar annulus results to higher-dimensional shells in both Dirichlet and Neumann cases, and provide a new family of Bessel-function estimates of independent interest. The long Bessel-function part (Sections 2-4) contains detailed, apparently sound proofs of uniform zero asymptotics (Theorems 2.20 and 4.13) and of real-simplicity of zeros (Theorem 3.3). The main weakness is the reliance of the improved exponent on Theorem 6.1, whose proof is only sketched and whose decisive input is the unpublished preprint [27] by one of the authors; this is a correctness-risk concern rather than an internal inconsistency.

major comments (2)
  1. [Section 6, Theorem 6.1] The improved exponent in Theorem 1.1 depends entirely on Theorem 6.1, but its proof is explicitly a sketch. The decisive assertion that condition (6.1) can be dropped for M in the range (6.2) is made in the final paragraph of the proof: it is stated that (6.1) is only needed in the case M ≳ T^{181/328} and that this is disjoint from (6.2), but the subcase analysis of Huxley's Lemmas 3.3 and 3.4 that would demonstrate that (6.1) is not used elsewhere is not displayed. Since [27] is an unreviewed preprint by one of the authors and its Theorem 4.2 is not reproduced, the reader cannot verify the key estimate (6.3). This is a load-bearing gap: if the nondegeneracy-free extension fails, the ball bound reverts to the Huxley exponent and the rational-shell improvement disappears. The authors should either give a complete proof of Theorem 6.1 or state the improved bounds as conditional on [27, Theorem 4.2].
  2. [Lemma 5.1, equations (5.9) and (5.16)] The application of Theorem 6.1 requires the functions F defined in (5.9) and (5.16) to satisfy |F^{(j)}(x)| ≍ 1 for j=1,2,3. For (5.16) the verification is deferred to "the size of derivatives of T (in [14, Lemma 4.7])", but the third-derivative bound is not shown, and the constant in the lower bound matters for the dyadic summation. Since this condition is part of the hypothesis of Theorem 6.1, the proof of Lemma 5.1 should explicitly verify it for both (5.9) and (5.16).
minor comments (5)
  1. [Introduction, page 3] The sentence "Even thought the planar domain for the corresponding lattice point problem could be bad" contains a typo: "thought" should be "though".
  2. [Theorem 5.4 and its proof] The leading coefficient is typeset as "2d(Γ(d/2 + 1))2"; this should be 2^d (Γ(d/2 + 1))^2 to avoid ambiguity.
  3. [References] Reference [27] is an arXiv preprint; it should be marked as such, with the version number and date, rather than appearing in the same format as journal articles.
  4. [Theorem 6.1] The summation ∑_{m=M}^{M_2} requires M to be an integer; either state that M is a positive integer or replace the lower limit by ⌊M⌋.
  5. [Remark 5.6] The discussion of Huxley's interpretation of [14] is a scholarly dispute rather than a mathematical result; consider moving it to a separate note or footnote so that the main text remains focused on the mathematical content.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the improved exponent 2θ* is an external exponential-sum input, not an eigenvalue fit.

full rationale

I find no circular step. The chain of derivation is structurally independent: eigenvalue counting for balls and shells is reduced, via Bessel zero approximations and multiplicity weights, to weighted planar lattice point counting (Sections 4 and 5). The main terms and remainder estimates are then obtained from Lemma 5.1, Lemma 5.3, Theorems 5.4 and 5.5, with the improved exponent entering only through Theorem 6.1, an estimate for rounding error sums. The constant θ* is not fitted to eigenvalue data; it is defined as the opposite of the fixed solution of equation (6.4), and the exponent 2θ* originates in [27, Theorem 4.2], a result about the Gauss circle and Dirichlet divisor problems, not about Weyl remainders. Thus the paper does not assume Theorem 1.1 to prove Theorem 1.1. The Bessel zero approximations in Sections 2 and 4 are derived from uniform asymptotics and are used as inputs, not as restatements of the target bounds. There is a genuine provenance and verification concern: Theorem 6.1, and specifically the claim that the nondegeneracy condition (6.1) is unnecessary in the range (6.2), is only sketched and is deferred to the coauthored preprint [27], whose proof is not reproduced or machine-checked. However, this is a correctness risk about an external exponential-sum estimate, not a circularity: the cited result is parameter-free, stated for a different problem, and does not include the target Weyl-law conclusion as an assumption. The paper's reduction from spectrum counting to lattice counting is independent of the bound being proved, and the exponent θ* is not chosen to match the final remainder. Accordingly, no step reduces to its own input by construction or by definition.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical constants are fitted to eigenvalue data. theta* solves the fixed equation (6.4) and originates in [27]. Most of the load-bearing analysis uses standard asymptotic and analytic number theory, but the strongest exponent relies on unpublished [27] and on the paper's own sketchy Theorem 6.1, which is not independently verified.

assumptions (6)
  • standard math Olver's uniform asymptotic expansions and Airy-function approximations for Bessel functions
    Used throughout Sections 2 to 5 as the basis for all zero asymptotics; quoted in Appendix A as equations (A.1) through (A.8), from [1] and [31].
  • domain assumption Cochran's theorem on the real simple zeros and zero counts of the cross-products f_nu and g_nu
    Invoked in Proposition 2.9, Proposition 2.10, and Section 3 to force exactly one zero per interval and to count zeros in large circles; cited as [5].
  • domain assumption Filonov, Levitin, Polterovich, and Sher phase-function lemmas for ultraspherical Bessel functions
    Used in Theorem 3.3 and Proposition 4.9 to show that the cross-product h_nu,delta has only real simple zeros and that j'_nu,delta has the expected number of zeros; cited as [12].
  • domain assumption Huxley's exponential sum estimates and second spacing lemmas
    Used in the proof of Theorem 6.1 and in the lattice counting arguments to estimate sawtooth sums; cited as [19].
  • domain assumption Li and Yang's [27, Theorem 4.2] exponential sum estimate for the Gauss circle and divisor problems
    The improved exponent 2theta* depends on this unpublished preprint by the last author; the current paper adapts it into Theorem 6.1 without reproducing the full proof.
  • domain assumption Bourgain and Watt's variant of the double large sieve inequality and Guth and Maldague's small cap decoupling for cones
    Used inside the proof sketch of Theorem 6.1, following [27], to reduce the exponential sum estimate to spacing problems; cited via [4] and [17].

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Cite this review

Pith. "Pith review of Bessel functions and Weyl's law for balls and spherical shells." pith.science (2026). https://pith.science/paper/STU32ZNQ

@misc{pith2026241214059,
  author       = {Pith},
  title        = {Pith review of: Bessel functions and Weyl's law for balls and spherical shells},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/STU32ZNQ}},
  note         = {Machine review of arXiv:2412.14059}
}
abstract

The purpose of this paper is twofold. One is to investigate the properties of the zeros of cross-products of Bessel functions or derivatives of ultraspherical Bessel functions, as well as the properties of the zeros of the derivative of the first-kind ultraspherical Bessel function. The properties we study include asymptotics (with uniform and nonuniform remainder estimates), upper and lower bounds and so on. In addition, we provide the number of zeros of a certain cross-product within a large circle and show that all its zeros are real and simple. These results may be of independent interest. The other is to investigate the Dirichlet/Neumann Laplacian on balls and spherical shells in $\mathbb{R}^d$ ($d\geq 2$) and the remainder of the associated Weyl's law. We obtain new upper bounds in all dimensions, both in the Dirichlet and Neumann cases. The proof relies on our studies of Bessel functions and the latest development in the Gauss circle problem, which was driven by the application of the emerging decoupling theory of harmonic analysis.

Figures

Figures reproduced from arXiv: 2412.14059 by the authors.

Figure 4.1
Figure 4.1. The domain Ω. In the Dirichlet case, we write NS(µ) = N D S (µ) = X∞ l=0 Ä md l − md l−1 ä N D l (µ), where N D l (µ) := #{(n, k) ∈ Z+ × N : ωn,k ≤ µ, n ≥ l}. The superscript “D” represents Dirichlet while “N” represents Neumann. Correspondingly in this case we define a weighted lattice point counting function (associated with the domain Ω) PΩ(µ) = PD Ω(µ) := X∞ l=0 Ä md l − md l−1 ä PD l (µ), where PD l (µ) := # {(… view at source ↗
Figure 4
Figure 4. ) because of [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 4.2
Figure 4.2. The domain Ω0. We define an analogous version corresponding to the Neumann Laplacian QΩ0 (µ) = QN Ω0 (µ) := X∞ l=0 Ä md l − md l−1 ä QN Ω0,l(µ), where QN Ω0,l(µ) := # {(ν, k + 1/4) ∈ µΩ0 : n ∈ Z+, n ≥ l, k ∈ Z+} . We have the following “ball” version of Proposition 4.2. Proposition 4.7. There exists a constant C > 0 such that |NB(µ) − QΩ0 (µ)| ≤ QΩ0 (µ + Cµ−3/7 ) − QΩ0 (µ − Cµ−3/7 ) + O Ä µ d−2+4/7 ä . The Dirichlet… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. P\'{o}lya's conjecture for higher-dimensional Neumann balls

    math.SP 2026-07 accept novelty 7.0 of 10

    For every d≥3 and every λ≥0, the Neumann counting function of the unit d-ball satisfies N(λ) ≥ w_d λ^d, so Pólya's conjecture holds for all Euclidean balls.

Reference graph

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