Paper Citation Record · LEDGER

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

As of 12 August 2026, this Paper Citation Record lists 36 of 36 outbound references and 1 inbound Pith citation observation for arXiv:2412.14059.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2412.14059 v1

Coverage vector

measured 36 of 36 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-11T12:38:32.073035Z

measured 37 of 37 standing notices

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Source: scholarly_work_events, retraction_status_cache, observed 2026-08-11T06:34:44.6726+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-03T09:43:11.897341Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

36 of 36 outbound references displayed

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verified fuzzy20
unresolved16
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External citation measurements

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Outbound references

Observation a52696c3-c79f-4fc2-b0f8-1e87c65066cc · outbound

This paper cites and Stegun, I.

Bessel functions and Weyl's law for balls and spherical shells and Stegun, I

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:33.635841Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 21adebd8-83fc-4e60-b53e-fb0b3427082a · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-11T12:38:33.592628Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation cf062a17-d2bf-40a4-8813-815dc78a9d7d · outbound

This paper cites Mean square of zeta function, circle problem and divisor problem revisited.

Bessel functions and Weyl's law for balls and spherical shells Mean square of zeta function, circle problem and divisor problem revisited

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-11T12:38:31.420316Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e4307250-d2a6-427b-b067-3b0886980a76 · outbound

This paper cites and Watt, N., Decoupling for perturbed cones and the mean square of |ζ( 1 2 + it)|, Int.

Bessel functions and Weyl's law for balls and spherical shells and Watt, N., Decoupling for perturbed cones and the mean square of |ζ( 1 2 + it)|, Int

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:33.572091Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 6c098285-ef54-4aeb-8c93-82c7b6b510a8 · outbound

This paper cites A., Remarks on the zeros of cross-product Bessel functions, J.

Bessel functions and Weyl's law for balls and spherical shells A., Remarks on the zeros of cross-product Bessel functions, J

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:33.552627Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 9e0fc006-9763-4f3c-918c-add63350721d · outbound

This paper cites A., The asymptotic nature of zeros of cross-product Bessel functions , Quart.

Bessel functions and Weyl's law for balls and spherical shells A., The asymptotic nature of zeros of cross-product Bessel functions , Quart

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:33.481315Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 39eb0e37-87a7-40f8-91f0-a8ffcd73668c · outbound

This paper cites A., The analyticity of cross-product Bessel function zeros , Proc.

Bessel functions and Weyl's law for balls and spherical shells A., The analyticity of cross-product Bessel function zeros , Proc

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-11T12:38:31.444055Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b517aa31-515b-49b1-b5c5-fcb7ecf58cc8 · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 8

Resolution: unresolved; raw_fallback, observed 2026-08-11T12:38:33.386688Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation cae6e0c4-21a0-4495-9a81-36c69a79d18d · outbound

This paper cites G., Zahlentheoretische Absch¨ atzungen mit Anwendung auf Gitter- punktprobleme (German), Math.

Bessel functions and Weyl's law for balls and spherical shells G., Zahlentheoretische Absch¨ atzungen mit Anwendung auf Gitter- punktprobleme (German), Math

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:33.306495Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation a1e3f71a-6553-40a0-991e-77a7ee86d098 · outbound

This paper cites and Toth, J.A., Smooth billiards with a large Weyl remainder, Int.

Bessel functions and Weyl's law for balls and spherical shells and Toth, J.A., Smooth billiards with a large Weyl remainder, Int

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:33.202163Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation ab13812e-11b7-4968-b912-9756479b28b8 · outbound

This paper cites and Sher, D., P´ olya’s conjecture for Euclidean balls, Invent.

Bessel functions and Weyl's law for balls and spherical shells and Sher, D., P´ olya’s conjecture for Euclidean balls, Invent

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:33.181724Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation d6d66c99-4e20-4e32-9b18-341f8e78bec8 · outbound

This paper cites and Sher, D., Uniform enclosures for the phase and zeros of Bessel functions and their derivative , SIAM J.

Bessel functions and Weyl's law for balls and spherical shells and Sher, D., Uniform enclosures for the phase and zeros of Bessel functions and their derivative , SIAM J

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:33.161220Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 970a1ad9-f5d1-4f61-8061-652265a5428b · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 13

Resolution: unresolved; raw_fallback, observed 2026-08-11T12:38:33.079664Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 0bfb147b-224c-44b4-b9df-ce4d8ad42dbe · outbound

This paper cites and Wang, Z., The Weyl formula for planar annuli , J.

Bessel functions and Weyl's law for balls and spherical shells and Wang, Z., The Weyl formula for planar annuli , J

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.969278Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation febd4823-21ca-44e7-87fd-6932d67bc650 · outbound

This paper cites and Wang, Z., An improved remainder estimate in the Weyl formula for the planar disk , J.

Bessel functions and Weyl's law for balls and spherical shells and Wang, Z., An improved remainder estimate in the Weyl formula for the planar disk , J

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.950118Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 703534cd-2acb-4c90-9b17-bc19cb389073 · outbound

This paper cites Introduction to harmonic analysis, group representations and applications , North-Holland Mathematics Studies, 174, North- Holland Publishing Co., Amsterdam, 1992.

Bessel functions and Weyl's law for balls and spherical shells Introduction to harmonic analysis, group representations and applications , North-Holland Mathematics Studies, 174, North- Holland Publishing Co., Amsterdam, 1992

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.931707Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 1c309ceb-c7d8-4b9c-8dbc-b61bb07ef2bd · outbound

This paper cites Amplitude dependent wave envelope estimates for the cone in $\mathbb{R}^3$.

Bessel functions and Weyl's law for balls and spherical shells Amplitude dependent wave envelope estimates for the cone in $\mathbb{R}^3$

Reference 17

Resolution: unresolved; no resolver link, observed 2026-08-11T12:38:31.497494Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 0305a5e3-94c9-4f25-bb3b-f4ab7d8f80ef · outbound

This paper cites E., Bessel phase functions: calculation and application , Numer.

Bessel functions and Weyl's law for balls and spherical shells E., Bessel phase functions: calculation and application , Numer

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.858716Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 8b1a64d2-a26e-484b-b8d5-66cc585cfb10 · outbound

This paper cites N., Exponential sums and lattice points.

Bessel functions and Weyl's law for balls and spherical shells N., Exponential sums and lattice points

Reference 19

Resolution: unresolved; no resolver link, observed 2026-08-11T12:38:31.665684Z

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Unavailable: canonical work link unavailable.

Observation 847abc8a-7084-40a7-940c-3e53ac5f395e · outbound

This paper cites N., Lattice points and Weyl’s formula for the disc , Acta Arith.

Bessel functions and Weyl's law for balls and spherical shells N., Lattice points and Weyl’s formula for the disc , Acta Arith

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.688778Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation f12b0117-2d46-4bc7-8989-c063dae169a0 · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 21

Resolution: unresolved; raw_fallback, observed 2026-08-11T12:38:32.614307Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 07150411-4bfe-4044-8173-127c3d668068 · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 22

Resolution: unresolved; raw_fallback, observed 2026-08-11T12:38:32.594344Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 6b89e323-47fe-460a-be18-9dcac292e8f8 · outbound

This paper cites V., Asymptotic formulae for eigenvalues of an elliptic membrane.

Bessel functions and Weyl's law for balls and spherical shells V., Asymptotic formulae for eigenvalues of an elliptic membrane

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.573050Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 8e2c7212-a5d5-464f-8a52-7b44220ef496 · outbound

This paper cites V., Asymptotic distribution of eigenfrequencies of a plane membrane in the case of separable variables.

Bessel functions and Weyl's law for balls and spherical shells V., Asymptotic distribution of eigenfrequencies of a plane membrane in the case of separable variables

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.555093Z

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Observation 1cd86dab-4407-41df-80f9-d4c445d38b5d · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 25

Resolution: unresolved; raw_fallback, observed 2026-08-11T12:38:32.529306Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation fb0f8384-814c-4883-8c77-13ceb77aee3e · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-11T12:38:32.015911Z

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Unavailable: canonical work link unavailable.

Observation 08ee2b8e-4ec1-4485-9e4c-7672299ee75f · outbound

This paper cites An improvement on Gauss's Circle Problem and Dirichlet's Divisor Problem.

Bessel functions and Weyl's law for balls and spherical shells An improvement on Gauss's Circle Problem and Dirichlet's Divisor Problem

Reference 27

Resolution: unresolved; no resolver link, observed 2026-08-11T12:38:32.022086Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e04b60c8-de17-4d9c-8fea-4c947e7a9c4f · outbound

This paper cites C., Lower bounds for the zeros of Bessel functions , Proc.

Bessel functions and Weyl's law for balls and spherical shells C., Lower bounds for the zeros of Bessel functions , Proc

Reference 28

Resolution: unresolved; no resolver link, observed 2026-08-11T12:38:32.028546Z

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Unavailable: canonical work link unavailable.

Observation 1cca504d-7300-4305-a091-9ce01b253f71 · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 29

Resolution: unresolved; no resolver link, observed 2026-08-11T12:38:32.033741Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 45944c53-b806-46ad-ba57-ef7d68edcf58 · outbound

This paper cites B., Weyl’s conjecture for manifolds with concave boundary , Geometry of the Laplace operator (Proc.

Bessel functions and Weyl's law for balls and spherical shells B., Weyl’s conjecture for manifolds with concave boundary , Geometry of the Laplace operator (Proc

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.398226Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 0682d097-6dde-4e84-a042-20d1ffdad70b · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 31

Resolution: unresolved; raw_fallback, observed 2026-08-11T12:38:32.377981Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation d8f3a50e-2c06-47b9-bed3-ed27f3058e52 · outbound

This paper cites an unresolved cited work.

Bessel functions and Weyl's law for balls and spherical shells Unresolved cited work

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-11T12:38:32.050235Z

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Unavailable: canonical work link unavailable.

Observation e2c4ddd9-fc21-46f6-8660-a1873219a174 · outbound

This paper cites and Yesha, N., Differences between Robin and Neumann eigenvalues, Comm.

Bessel functions and Weyl's law for balls and spherical shells and Yesha, N., Differences between Robin and Neumann eigenvalues, Comm

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.346352Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation 3888eb4f-a496-4be0-ba37-961a3d6b4e6f · outbound

This paper cites N., A treatise on the theory of Bessel functions , Reprint of the second (1944) edition, Cambridge Mathematical Library, Cambridge University Press, Cam- bridge, 1995.

Bessel functions and Weyl's law for balls and spherical shells N., A treatise on the theory of Bessel functions , Reprint of the second (1944) edition, Cambridge Mathematical Library, Cambridge University Press, Cam- bridge, 1995

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.285629Z

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No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.

Observation e2693a00-b04d-4a52-9563-dfb82aca52f1 · outbound

This paper cites (German), Math.

Bessel functions and Weyl's law for balls and spherical shells (German), Math

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.211747Z

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Observation 1f2a9cee-772e-4517-8f12-b827fe2c7700 · outbound

This paper cites (German), J.

Bessel functions and Weyl's law for balls and spherical shells (German), J

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-11T12:38:32.170320Z

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Pith citing papers

Observation 611074a1-2e2e-4b06-94d7-c64f643808ac · inbound

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

P\'{o}lya's conjecture for higher-dimensional Neumann balls Bessel functions and Weyl's law for balls and spherical shells

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-03T09:43:11.897341Z

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Unavailable: canonical work link unavailable.