Generalized Hardy's identity for the astroid-type p-circle lattice point problem

Masaya Kitajima

REVIEW 3 major objections 5 minor 17 references

Generalized Hardy's identity for the astroid-type p-circle lattice point problem

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves an exact identity expressing the lattice-point discrepancy of astroid-type p-circles as an integral of generalized Bessel functions.

desk verdict The differential formula is solid, but the main identity is formally derived and fails for p=1; the theorem needs a major rewrite. read the letter →

arxiv 2506.03331 v2 pith:HQDVQQSI submitted 2025-06-03 math.NT

Masaya Kitajima This is my paper

classification math.NT MSC 11P2142B0533C1026A33

keywords latticepointproblemp-circleLamécurveHardy'sidentitygeneralizedBesselfunctionsErdélyi-KoberoperatorPoissonsummationformulaGausscircle

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

For the plane curves $|x_1|^p + |x_2|^p = r^p$, with $2/p$ a positive integer, this paper derives an exact expression for the discrepancy $P_p(r)$ between the number of lattice points inside the curve and its area. The expression writes $P_p(r)$ as an integral over scaled radii of a sum of generalized Bessel functions, with the sum taken over lattice points on the curve's boundary. At $p=2$ the formula reduces to Hardy's identity, the classical series that motivated Hardy's conjectured optimal bound for the circle problem. The paper's aim is to supply the same kind of analytic starting point for the astroid-type cases, where no such identity had been available.

What carries the argument

The machinery is the family of generalized Bessel functions $J^{[p]}_\omega$ and their distorted-angle components $J^{[p]}_{\omega,\varphi}$, defined by a series involving Gamma factors and a polynomial $\Phi^{[p]}_{k,\varphi}$ in $\cos^{4/p}\varphi$ and $\sin^{4/p}\varphi$. The load-bearing identity is the differential formula $$\frac{d}{dr}\, $r^{{1+(p-1)\omega}}$ $J^{{[p]}}$_{\omega+1,\varphi}(r) = $r^{{1+(p-1)\omega}}$ $J^{{[p]}}$_{\omega,\varphi}(r),$$ which generalizes the classical Bessel recurrence and converts the radial integral appearing after Poisson summation into the boundary term in Theorem 1.2. The Erdélyi-Kober fractional integral and derivative are the tools used to discover and prove this formula.

What would settle it

For a concrete admissible value such as $p=2/3$, compute $P_p(r)$ directly from lattice counts at integer radii and compare with a high-precision numerical evaluation of the right-hand side of (2.15), truncated at a large radius $S$; any systematic mismatch beyond quadrature error would indicate that the Poisson summation step fails.

Watch

Extended reading notes

Core claim

Theorem 1.2 asserts that for $p>0$ with $2/p\in\mathbb{N}$, the lattice point error satisfies $$P_p(r)=\frac{p\Gamma(1/p)^2}{2\pi}\, r \int_1^\infty \frac{1}{$s^{{1/p}}$}\left(\sum_{\varphi\in $A_s^{{[p]}}$} $J^{{[p]}}$_{1,\varphi}(2\pi $s^{{1/p}}$r)\right)d\mu(s),$$ where $A_s^{[p]}$ is the finite set of distorted angles belonging to lattice points on the $p$-circle of radius $s^{1/p}$, and $J^{[p]}_{\omega,\varphi}$ are the distorted-angle components of the generalized Bessel functions introduced by the author. The identity is exact, not a formal expansion, and when $p=2$ the angle dependence disappears and the right-hand side becomes Hardy's identity with $J_1$ and the representation-number function $R(k)$. The proof routes a Poisson summation formula through the generalized Hankel transform and uses a differential formula for $J^{[p]}_{\omega,\varphi}$ to evaluate the radial integral. The author notes that convergence of the integral has been confirmed only conditionally for $p=2$; absolute convergence for general admissible $p$ is left open.

Load-bearing premise

The proof applies Poisson summation to the indicator function of the $p$-ball without first checking that the resulting lattice sum converges absolutely; if that conversion cannot be justified by a limiting or analytic-continuation argument, the identity in Theorem 1.2 is not established.

Editorial extensions

If this is right

For every $p$ with $2/p\in\mathbb{N}$, the discrepancy $P_p(r)$ is now written as one exact integral formula, giving a starting point for asymptotic analysis of cases that previously had no identity.
At $p=2$ the formula reproduces Hardy's identity, so the classical circle discrepancy expression is a special case of the new identity.
The author conjectures that if a uniform asymptotic bound $J^{[p]}_{1,\varphi}(r)=O(r^{-q})$ can be proved, the identity would imply the optimal estimate $P_p(r)=O(r^{1-q+\varepsilon})$ for every small $\varepsilon>0$.
The differential formula (2.13) connects generalized Bessel functions to Erdélyi-Kober fractional calculus, pointing toward fractional differential equations satisfied by $J^{[p]}_{\omega,\varphi}$.

Reading between the lines

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

The editor's reading: if the Poisson summation step can be justified by a limiting argument on truncated balls, the same identity should extend beyond the $2/p\in\mathbb{N}$ family, since the formal mechanism does not obviously require that restriction.
An extension not pursued here: a direct numerical check for a concrete value such as $p=2/3$ would be a cheap test of whether the conditional-convergence issue actually changes the value of the right-hand side.
The same distorted-angle decomposition could be applied to the $p$-ellipse family $\{s x_1^p + x_2^p/s = r^p\}$, whose lattice-point extremal problem has already attracted attention.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper extends Hardy's identity to p-circles with 2/p in N. The genuinely new piece is the differential formula (Prop 2.3) for the author's generalized Bessel functions, derived via Erdélyi-Kober operators and checked by series. That part looks right; the p=2 case reduces to the classical Bessel recurrence. The identity (2.15) is a clever formal calculation, and its reduction to Hardy for p=2 is correct.

The load-bearing step is Poisson summation applied to a discontinuous indicator. Stein-Weiss Theorem 2.4 requires hypotheses not satisfied here: the Fourier coefficients are not absolutely summable. The author himself concedes convergence of (2.15) is unconfirmed except conditional for p=2. This is not a technicality: for p=1, integer r, all nonzero Fourier coefficients of the indicator vanish, so the RHS of (2.15) is 0 while the true discrepancy is 1-2r. So Theorem 1.2 as stated is false. The counterexample uses the same change of variables as Section 2.3, so the flaw is in the theorem statement, not just the proof.

The Erdélyi-Kober computation has a slip or inconsistency, but it is superseded by the series proof in Remark 2.4, so that is minor. The paper is transparent about the gap in the concluding remarks, which helps the reader, but the theorem is stated unconditionally. The generalized Bessel functions and the differential formula may be salvageable and useful for future work. The paper is not ready for publication as is. A serious referee would be needed to untangle the hypotheses under which a corrected identity might hold—for example, non-integer r, a boundary convention, or a summability method.

If this comes across my desk, I would not desk-reject it outright, but I would send it to a referee familiar with Poisson summation and lattice point problems, expecting a major revision or a restriction of the theorem. The differential formula alone might warrant a short note, but the main theorem as written is false.

Referee Report

3 major / 5 minor

Summary. This paper considers the lattice point discrepancy P_p(r)=N_p(r)-(2/p)Γ(1/p)^2/Γ(2/p) r^2 for the astroid-type p-circle |x_1|^p+|x_2|^p=r^p, for p with 2/p∈N. The author introduces generalized Bessel functions J^{[p]}_{ω,φ}, proves a decreasing-order differential formula, and states Theorem 1.2, a generalized Hardy identity expressing P_p(r) as an integral, with respect to a counting measure, against the sum of J^{[p]}_{1,φ}(2π s^{1/p} r) over distorted angles. The proof applies Poisson summation to the indicator of the p-ball and then changes to distorted polar coordinates. The paper also connects the differential formula to Erdélyi-Kober fractional operators and to the author's earlier results.

Significance. If the main identity were correct, it would be a natural extension of Hardy's identity and a possible starting point for discrepancy estimates in the previously untreated range 0<p<2. The paper has genuine strengths: the series proof of the differential formula in Remark 2.4 is explicit and checkable, the p=2 case correctly reduces to Hardy's identity, and the connection with Erdélyi-Kober operators is interesting. However, the central identity is not established. The Poisson summation step is applied to a sharp indicator without verifying pointwise convergence, the convergence of the series in (2.15) is explicitly left open in Section 3, and the identity is in fact false as stated for p=1 at integer radii. The announced main theorem therefore cannot be accepted in its current form.

major comments (3)

[Theorem 1.2 and §2.3, Eq. (2.15)] The main theorem is false as stated. Take p=1, which is allowed because 2/p=2∈N, and let r be a positive integer. For F_r=1_{|x_1|+|x_2|<r}, the Fourier transform at n=(n_1,n_2)∈Z^2\{0} is 2 sin(π r(n_1+n_2)) sin(π r(n_1-n_2))/(π^2(n_1+n_2)(n_1-n_2)), with the usual limiting interpretation at zeros, and this vanishes for every nonzero n when r is an integer. Following the paper's own change of variables in §2.3, this forces J^{[1]}_{1,φ}(2π s r)=0 for every s in the support of the counting measure, so the right-hand side of (2.15) is 0. The actual strict count is N_1(r)=2r(r-1)+1, the area is 2r^2, and therefore P_1(r)=1-2r, which is nonzero for every r≥1. For example, at r=1, P_1(1)=-1 while the formula gives 0. Thus Theorem 1.2 cannot hold for all positive r as claimed.
[§2.3, Poisson summation step] The proof invokes the Poisson summation formula for the periodization of the integrable function F_r (Stein-Weiss Theorem 2.4) and then evaluates the resulting equality at x=0. That theorem gives an L^1 equality of periodizations; it does not justify pointwise evaluation at x=0 unless the Fourier series of the periodization converges at that point. Here \hat F_r(n)=O(|n|^{-2}) in general, so the Fourier series is not absolutely convergent. The p=1 integer-r example shows that pointwise evaluation can indeed fail: the Fourier series collapses to the constant area term rather than to N_1(r). A limiting or analytic-continuation argument is required but not supplied. The paper itself concedes in Section 3 that the convergence of (2.15) has not been confirmed except for conditional convergence when p=2.
[Theorem 1.2, statement] The theorem does not specify the boundary counting convention. In §2.3, N_p(r) counts lattice points satisfying the strict inequality |x|_p<r, while the boundary of the p-circle can contain lattice points for exceptional radii, for example p=1 and integer r. The right-hand side of (2.15) is sensitive to this choice, as the p=1 example demonstrates. At minimum, the theorem must state the counting convention and either exclude the exceptional radii or prove a version that handles them.

minor comments (5)

[Introduction, first paragraph] The phrase "supper ellipse" should be "superellipse".
[Introduction, third paragraph] The sentence "the following important theorem by E. Krätzel is given by the decomposition" is awkward; "by the decomposition" should be "via the decomposition".
[Remark 2.2] The phrase "Erdélyi-Kober type fractional integrals operator of multiple variables" should be "operators of multiple variables".
[Equations (1.3) and (1.4)] The multi-index notation N^2_0 is used in (1.3) before it is defined in the sentence following (1.4); the definition should appear at first use.
[§2.3, Eq. (2.15)] The integral with respect to the counting measure dµ(s) should be written as a sum over the discrete set of possible values of s=|n|_p^p, or the notation should explicitly identify that set; otherwise the expression is ambiguous.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the generalized Hardy identity is a Poisson-summation transform, not a fitted or definitional reduction; the main weakness is analytic validity, not circularity.

full rationale

The derivation of Theorem 1.2 is not circular. The right-hand side of (2.15) is obtained by an explicit Poisson-summation calculation: the indicator of the p-ball is transformed using the p-Hankel kernel J_p^[p], the nonzero lattice sum is grouped by the angle sets A_s^[p], and Proposition 2.3's differential formula converts the integral of tau J_0,phi^[p] into r J_1,phi^[p]. The generalized Bessel functions are introduced from the author's earlier work [9], but in this paper their uniform convergence (Proposition 2.1) and the differential formula (Proposition 2.3) are proved directly, and no parameter is fitted to the lattice-point discrepancy P_p(r). For p=2, the identity reduces to Hardy's known identity, which is an external consistency check. The citations to [9] and [10] supply the p-Hankel transform and prior asymptotic results, but those results are parameter-free, stated with assumptions, and do not contain the target identity as a hidden input, so they do not make the argument circular. The paper itself admits a real limitation in Section 3: the convergence of (2.15) is not confirmed except conditionally for p=2. Moreover, the Poisson-summation step in Section 2.3 is applied to the indicator without fully verifying the hypotheses, and the p=1 case gives a concrete counterexample to the stated theorem. These are foundational correctness gaps, not circular reductions: the claimed identity is not equivalent to its inputs by construction, but rather is potentially invalid for some admitted p. Accordingly, the circularity score is low, while correctness risk is high.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central identity depends on a chain of tools: the author's generalized Bessel functions, the generalized Hankel transform from [9], and Stein-Weiss Poisson summation. No data or fitted constants appear. The main burden is the analytic justification of Poisson summation and the conditional convergence of the final series, which is the paper's weakest point.

assumptions (3)

standard math Poisson summation formula for integrable functions (Stein-Weiss, Theorem 2.4)
Used in Section 2.3 to convert the sum over lattice points of the indicator function into a sum of its Fourier transforms. The paper does not verify the theorem's hypotheses for the sharp indicator of the p-ball; the resulting series is not absolutely convergent.
ad hoc to paper Generalized Hankel transform representation for p-radial functions from [9]
The formula for the Fourier transform of a p-radial function in terms of J^[p]_0 is cited from the author's own earlier paper [9]. It is load-bearing for the Poisson summation step and is not re-proved here.
standard math Erdelyi-Kober fractional integral/derivative properties ([8], (2.6.1), (2.6.29), (2.6.43))
Used in Section 2.2 to prove the differential formula and the inversion identity (2.12). These are standard textbook results in fractional calculus; however, the displayed computation contains exponent mismatches as printed.

invented entities (1)

Generalized Bessel functions J^[p]_omega and J^[p]_{omega,phi}
purpose: Serve as the kernel in the p-radial Fourier/Hankel transform and in the generalized Hardy identity.
Introduced by the author in [9] and adapted here with a distorted angle coordinate. They reduce to standard Bessel functions only at p=2; no independent external evidence is provided beyond that calibration.

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Pith. "Pith review of Generalized Hardy's identity for the astroid-type p-circle lattice point problem." pith.science (2026). https://pith.science/paper/HQDVQQSI

@misc{pith2026250603331,
  author       = {Pith},
  title        = {Pith review of: Generalized Hardy's identity for the astroid-type p-circle lattice point problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HQDVQQSI}},
  note         = {Machine review of arXiv:2506.03331}
}

abstract

Let $r$ be a positive real number and $p$ satisfy $(2/p)\in\mathbb{N}$. Then, we consider the lattice point problem of the closed curves astroid-type $p$-circle $\{x\in\mathbb{R}^{2}|\ |x_{1}|^{p}+|x_{2}|^{p}=r^{p}\}$ which generalize the circle. In investigating the asymptotic behavior of the error term in the area approximation of the circle, G.H. Hardy conjectured an infimum for the evaluation in 1917. One of the grounds for this conjecture is the Hardy's identity, which is a series representation of the term, consisting of the Bessel function of order one and a certain number-theoretic function. In order to investigate an infimum in the error evaluation of the astroid-type $p$-circle, which is unknown in previous studies, in this paper, we derive generalized Hardy's identity for the figures by using generalized Bessel functions. Furthermore, the differential formula for the functions, which is important for the proof of this identity, is closely related to the Erd\'{e}lyi-Kober operator, and this formula and operator are expected to be useful in our future research.

Figures

Figures reproduced from arXiv: 2506.03331 by the authors.

**Figure 1.** Examples of the p-circle and the approximation by unit squares. is 1 2 -order (Hardy’s conjecture), based on his previous results[4] which established P2(r) ̸= O(r 1 2 ), P2(r) = Ω(r 1 2 (log r) 1/4 ) and Hardy’s identity (for example, see also [11], Theorem 3.12) P2(r) = r X∞ k=1 R(k) k 1 2 J1(2πk 1 2 r), (1.2) with Jω: the Bessel function of order ω, R(k) := #{n ∈ Z 2 | |n| 2 = k}. Since then, many mathematicians … view at source ↗

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Works this paper leans on

17 extracted references · 10 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Aomoto, K

K. Aomoto, K. Iguchi, On quasi-hypergeometric functions, Methods and Appli. of Analysis, 6 (1999) 55-66, https://doi.org/10.4310/MAA.1999.v6.n1.a4

work page doi:10.4310/maa.1999.v6.n1.a4 1999

[2] [2]

Gauss, De nexu inter multitudinem classium, in quas formae binariae secundi gradus distribuuntur, earumque determinantem, In: Schering, E., ed., Werke, Vol

C.F. Gauss, De nexu inter multitudinem classium, in quas formae binariae secundi gradus distribuuntur, earumque determinantem, In: Schering, E., ed., Werke, Vol. 2. G¨ ottingen: K¨ oniglichen Gesellschaft der Wissenschaften (1876) 269-291, https://doi.org/10.1017/ CBO9781139058230.012

work page

[3] [3]

Hankel, Die Cylinderfunctionen erster und zweiter Art, Math

H. Hankel, Die Cylinderfunctionen erster und zweiter Art, Math. Ann.1 (1869) 467-501, https: //doi.org/10.1007/BF01445870. 12

work page doi:10.1007/bf01445870

[4] [4]

Hardy, On the expression of a number as the sum of two squares, Q

G.H. Hardy, On the expression of a number as the sum of two squares, Q. J. Math. 46 (1915) 263-283, http://refhub.elsevier.com/S0022-1236(21)00354-2/ bibE0749EAC038C473558FC31E2A34CC3E5s1

work page 1915

[5] [5]

Hardy, E

G.H. Hardy, E. Landau, The average order of the arithmetical functions P (x) and ∆(x), Proc. Lond. Math. Soc. 15 (1917) 192-213, https://doi.org/10.1112/plms/s2-15.1.192

doi:10.1112/plms/s2-15.1.192 1917

[6] [6]

Huxley, Exponential sums and lattice points

M.N. Huxley, Exponential sums and lattice points. III, Proc. Lond. Math. Soc. (3) 87(3) (2003) 591-609, https://doi.org/10.1112/S0024611503014485

doi:10.1112/s0024611503014485 2003

[7] [7]

Ivi´ c, E

A. Ivi´ c, E. Kr¨ atzel, M. K¨ uhleitner, W.G. Nowak, Lattice points in large regions and re- lated arithmetic functions: Recent developments in a very classic topic, arXiv:math/0410522v1 (2004), https://doi.org/10.48550/arXiv.math/0410522

arXiv doi:10.48550/arxiv.math/0410522 2004

[8] [8]

Kilbas, H.M

A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier. North-Holland mathematics studies. 204 (2006)

work page 2006