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REVIEW 4 major objections 4 minor 39 references

Diophantine FLINT-HILLS series

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Pith's one-line read The paper claims to prove that the Flint-Hills series converges to approximately 30.314510, and that the irrationality measure of π is at most 5/2.

desk verdict Claims convergence of the Flint-Hills series and mu(pi) ≤ 2.5, but the proof rests on false regularity assumptions and circular tail estimates. read the letter →

arxiv 2502.03474 v1 pith:OBMNSGZN submitted 2025-01-21 math.GM

Nikos Mantzakouras , Carlos López Zapata This is my paper · ORCID
classification math.GM MSC 40A0511J82
keywords Flint-HillsseriesDiophantineDirichletRiemann-StieltjesintegralirrationalitymeasureHöldercontinuitymodifiedBesselfunctionsFermi-Diracpolygammafunction
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 claims to settle the long-standing question of whether the Flint-Hills series $\sum_{n=1}^{\infty} \csc^2(n)/n^3$ converges. The authors assert that it does, converging to $(4/3)\zeta(3) + (2\sqrt{3}/(3\pi))c_1 \approx 30.314510$, where $c_1 \approx 78.1160806386$ is defined through an infinite sum of ratios of modified Bessel functions. Because a known result [2] says convergence implies that the irrationality measure of $\pi$ is at most $5/2$, and a converse result [18] supplies the other direction, the paper concludes $\mu(\pi) \le 5/2$, improving the previous upper bound of about 7.6063. Three supporting routes are presented: a recursive trigonometric substitution, an asymptotic analysis of modified Bessel functions, and a Riemann-Stieltjes integral representation justified by Young's Hölder-continuity criterion.

What carries the argument

The central object is the Riemann-Stieltjes integral representation $\int_1^{\infty} (\csc^2(x)/x^3)\,d\lfloor x\rfloor$, which reproduces the Flint-Hills series because the floor function jumps by 1 at each integer. The paper argues the integral is well-defined through Young's criterion: if the integrand is $\alpha$-Hölder continuous and the integrator is $\beta$-Hölder continuous with $\alpha+\beta > 1$, the Riemann-Stieltjes integral exists. The evaluation machinery is built on the triple-angle identity for the cosecant, which rewrites $\csc^2(n)$ in terms of ratios of Bessel functions, and on the asymptotic expansion of the modified Bessel function $I_{1/2}(z)$; these ingredients produce the constant $c_1$ and the closed form $(4/3)\zeta(3) + (2\sqrt{3}/(3\pi))c_1$. A secondary mechanism is the function $\Lambda(t)$, defined through modified Bessel ratios, whose derivatives are expressed through polygamma functions, giving the partial-sum formula and the double-sided inequalities that bound the series.

What would settle it

Test the Hölder premise directly: for $x = \pi - \varepsilon$ and $y = \pi + \varepsilon$, the quotient $|\csc^2(x)/x^3 - \csc^2(y)/y^3|/|x-y|$ is of order $\varepsilon^{-2}$ as $\varepsilon \to 0$, so no finite Hölder constant $C$ can exist; this observation decides whether the proof's key premise holds. Separately, compute the partial sums $\sum_{n=1}^{N} \csc^2(n)/n^3$ for increasing $N$ and compare with $30.314510$ to test the claimed value.

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Extended reading notes

Core claim

The paper's central claim, stated on its own terms, is that the Flint-Hills series $\sum_{n=1}^{\infty} 1/(n^3 \sin^2 n)$ converges to the value $(4/3)\zeta(3) + (2\sqrt{3}/(3\pi)) c_1 \approx 30.314510$, where $c_1$ is the limit of the modified-Bessel-function sum $\Lambda(\sigma) = -i \sum_{n=1}^{\sigma-1} I_{1/2}(-3in)/(n^4 I_{1/2}(-in)^3)$. The derivation works by writing the series as the Riemann-Stieltjes integral $\int_1^{\infty} (\csc^2(x)/x^3)\,d\lfloor x\rfloor$ and invoking Young's criterion that such an integral is well-defined when the integrand is $\alpha$-Hölder and the integrator $\beta$-Hölder with $\alpha+\beta > 1$. The asymptotic behaviour of the modified Bessel functions is used to evaluate the tail of the series and to pin down the constant $c_1$, while the floor-function integrator turns the integral back into the original series term by term. From the claimed convergence, the paper derives that the irrationality measure of $\pi$ satisfies $\mu(\pi) \le 5/2$, citing the implication in [2] and the near-converse in [18].

Load-bearing premise

The argument's load-bearing premise is that the integrand $\csc^2(x)/x^3$ and the integrator $\lfloor x\rfloor$ satisfy Hölder-type bounds with exponents $\alpha$ and $\beta$ whose sum exceeds 1 on the whole half-line $[1,\infty)$.

Editorial extensions

If this is right

  • If the claimed convergence is correct, the upper bound on the irrationality measure of $\pi$ drops from the previous $\approx 7.6063$ to $5/2$.
  • The same Riemann-Stieltjes framework is claimed to prove convergence of the Cookson-Hills series $\sum \sec^2(n)/n^3$ and of $\sum \cot^2(n)/n^3$, with the latter equal to the Flint-Hills value minus $\zeta(3)$, about $29.11204$.
  • The closed-form value $30.314510$ is a concrete numerical target for partial-sum computations, and the paper supplies double-sided polygamma bounds such as $30.2842 \lesssim \Psi(\sigma) \lesssim 30.2920$ at $\sigma = 10001$.
  • The Hölder-inequality connection yields a family of bounds of the form $\sum \csc^2(n)/n^3 \lesssim \pi^2/(6\delta^2)$ with $\delta \approx 0.2329$, giving explicit control of the series in terms of a single parameter.

Reading between the lines

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

  • If the Hölder-regularity premise behind the Riemann-Stieltjes proof fails, the paper's demonstration of convergence via Young's criterion would not be valid, even if the numerical value is right; the convergence question would revert to an open problem.
  • The rapid decay of the Bessel-ratio terms suggests the claimed value could be verified to many digits by direct summation of the original Flint-Hills partial sums, so a persistent mismatch with $30.314510$ would be decisive.
  • Applying the same closed-form evaluation to the shifted series $\sum \csc^2(n+\pi/2)/n^3$ would yield an explicit numerical prediction for the Cookson-Hills series, going beyond the paper's remark that its integer part is 42.
  • The Fermi-Dirac and Bose-Einstein inequalities open a statistical reading of the series in terms of how often $n$ comes close to a multiple of $\pi$; the paper does not quantify this connection.
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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

4 major / 4 minor

Summary. The paper claims to prove convergence of the Flint-Hills series ∑_{n=1}^∞ csc^2(n)/n^3, with the value (4/3)ζ(3) + (2√3/(3π))c_1 ≈ 30.314510 for a constant c_1 ≈ 78.1160806386, and consequently an upper bound μ(π) ≤ 5/2 on the irrationality measure of π. The proof is attempted through several routes: an asymptotic analysis of modified Bessel functions leading to a partial-summation formula, a Riemann-Stieltjes integral approach using Hölder continuity and Young's criterion, and Hölder inequalities connected to Fermi-Dirac and Bose-Einstein integrals. A final section proposes a framework involving Weierstrass elliptic curves and polygamma functions for the same series.

Significance. If correct, the paper would resolve a long-standing open problem in number theory and substantially improve the best known upper bound for μ(π). The authors correctly identify the relevant background literature, including Alekseyev's and Meiburg's results connecting the Flint-Hills series to irrationality measures, and the numerical Bessel-function representation of the partial sums is suggestive. However, the central mathematical arguments contain multiple false premises and circular steps, and the claimed convergence is not established. Because the main conclusions rest on these unsupported steps, the paper is not publishable in its present form.

major comments (4)
  1. [§2.2.3, Theorem 2.11] The Hölder-continuity premise of the theorem is false. The function f(x)=csc^2(x)/x^3 has poles at every multiple of π; for x=π+ε and y=π−ε, |f(x)−f(y)| ∼ 2π^{-3}ε^{-2}, so |f(x)−f(y)|/|x−y| ∼ π^{-3}ε^{-3}→∞ as ε→0. Thus f is not 1-Hölder continuous on any interval containing a multiple of π. The claim that g(x)=⌊x⌋ is β-Hölder continuous with β=1 is also false: |⌊0.9⌋−⌊1.1⌋|=1>0.2=|0.9−1.1|. Young's criterion therefore cannot be invoked, and the conclusion that the Riemann-Stieltjes integral is well-defined and convergent is unsupported.
  2. [§2.1.2, Eqs. (2.19)–(2.24)] The derivation of the partial-summation formula is circular. Equation (2.19) expresses the asymptotic tail Θ of the Bessel series in terms of the unknown tail ∑_{n=σ}^∞ csc^2(n)/n^3. Equation (2.23) then contains this full tail on the right-hand side, and in (2.24) the infinite series ∑_{n=1}^∞ csc^2(n)/n^3 is subtracted from itself to obtain the partial sum. This subtraction is legitimate only if the infinite series is already known to converge, which is exactly the claim being proved. The argument assumes the convergence it seeks to establish.
  3. [§2.4, Theorem 2.13] The proof states that the series converges because 'csc^2(n) is bounded except at poles, and n^{-3} decays rapidly enough to ensure convergence by comparison to the p-series'. This is incorrect: csc^2(n) is not bounded on the positive integers. Since π is irrational, for the convergents p/q of π we have |sin p| ≤ |p−qπ| < 1/q, so csc^2(p) ≥ q^2, and the terms can be much larger than 1/p^3. The comparison with ∑1/n^3 therefore fails, and the proof does not establish convergence.
  4. [§2.8, Eqs. (2.33)–(2.35)] The Hölder-inequality argument assumes a uniform lower bound |sin k| ≥ δ for all large k to bound ∑1/(k sin k)^2 by ζ(2)/δ^2. No such positive uniform δ exists because the sequence |sin k| has 0 as a limit point. The bound in (2.34) and the subsequent estimate ∑ csc^2(n)/n^3 ≲ π^2/(6δ^2) are therefore unsupported.
minor comments (4)
  1. [§1.3, Theorem 1.2] The proof asserts that α-Hölder and β-Hölder conditions imply bounded variation; this is not true in general. The theorem and its proof should be corrected or rephrased in terms of bounded p-variation, which is the hypothesis actually needed for Young's criterion.
  2. [§1.1, Eq. (1.34)] The limiting step in (1.34) is internally inconsistent: L is first set equal to 1/constant and then treated as a variable tending to infinity. The same symbol cannot serve both as a fixed parameter and as the limit variable in a single derivation.
  3. [§2.1.3] The claimed rapid decay of I_{1/2}(−i3n)/I_{1/2}(−in)^3 is not rigorously demonstrated; the displayed asymptotic expression contains oscillatory exponential factors, and the argument that it decays sufficiently rapidly to make Λ(σ) converge needs a precise estimate rather than a heuristic assertion.
  4. [§2.4, Theorem 2.12] In the proof of Theorem 2.12, the statement that irrationality of α and β ensures ϕ(α,β)·t is never a rational multiple of π is not justified: ϕ is defined only as a multilinear function with ϕ(α,β)=f(πα,πβ), and the claimed avoidance of poles requires an explicit hypothesis on ϕ.

Circularity Check

3 steps flagged · score 8.0 of 10

The central 'proof' cancels the unknown Flint-Hills tail against itself and derives the constant c1 from the assumed constancy of the partial sums; the Hölder/Young route assumes the improper-integral convergence it is meant to establish.

  1. self definitional [Section 2.1.1–2.1.2, eqs. (2.19)–(2.24)]
    "Θ ∼ = ( −2π√3/3 ) ( ∑_{n=σ}^∞ 1/n^3 − 3/4 ∑_{n=σ}^∞ csc^2(n)/n^3 ) ... ∞∑_{n=1} csc^2(n)/n^3 ∼ = 4/3 ζ(3) + 2√3/3π (Λ + π√3/3 ψ′′(σ) + π√3/2 ∞∑_{n=σ} csc^2(n)/n^3 ) ... ∞∑_{n=1} csc^2(n)/n^3 − ∞∑_{n=σ} csc^2(n)/n^3 = ∑_{n=1}^{σ−1} 1/(n^3 sin^2(n)) ∼ = 4/3 ζ(3) + 2√3/3π Λ(σ) + 2/3 ψ′′(t)|_{t=σ}."

    The tail Θ of the Bessel-series split is asymptotically rewritten as a combination of ∑_{n=σ}^∞ 1/n^3 and the Flint-Hills tail ∑_{n=σ}^∞ csc^2(n)/n^3 itself. Substituting this Θ into the full series and then subtracting the same Flint-Hills tail from both sides treats that tail as a finite number that can be canceled. Finiteness of the tail is precisely the convergence of the Flint-Hills series that the paper claims to prove; the resulting partial-sum identity is therefore assumed rather than derived.

  2. self definitional [Section 2.1.4, Corollary 2.4, and eq. (1.6)]
    "Since the first derivative of a constant value approached by the partial sum of the Flint-Hills series ... i.e., d/dt(Sσ) = 0, it is expected to obtain the same consistency when Ψ′(t)|_{t=σ} = 0. ... c1 = lim_{σ→∞} ( −i∑_{n=1}^{σ−1} I1/2(−3i·n)/(n^4 I^3_{1/2}(−i·n)) ) ≈ 78.1160806386."

    The premise that Sσ is constant is exactly the convergence of the Flint-Hills partial sums, which is the conclusion to be established. From that premise the corollary derives Λ(t)=c1−π/√3 ψ''(t), and c1 is then numerically identified with the limit of the Bessel series. The final claimed value is assembled from this c1, so the 'prediction' is computed from an assumed constant limit rather than proved.

1 more flagged steps
  1. other [Section 2.2.3, Theorem 2.11]
    "The function f(x) = csc2(x)/x3 is α-Hölder continuous with α = 1, and g(x) = [x] is β-Hölder continuous with β = 1. Since α + β = 2 > 1, the Riemann-Stieltjes integral ∫∞1 csc2(x)/x3 d[x] is well-defined and convergent by Young's criterion for Stieltjes integration. Thus, the series converges."

    Young's criterion controls finite-interval Stieltjes integrals under α+β>1; it supplies no bound for the improper integral over [1,∞), whose tail convergence is equivalent to the Flint-Hills series itself. Invoking it to conclude the improper integral is 'well-defined and convergent' presupposes the sought convergence. The stated Hölder premises are also false: csc2(x)/x3 has poles at every multiple of π, so it is not 1-Hölder, and ⌊x⌋ jumps by 1 at integers, so |⌊x⌋−⌊y⌋|≤|x−y| fails.

full rationale

The paper's central convergence claim is not derived from an independent estimate. The Bessel-function route rewrites the unknown Flint-Hills tail in terms of itself and cancels it from both sides; the partial-sum formula (1.1)/(2.24) is exactly the convergence assumption recycled. The constant c1, which fixes the numerical value 30.314510, is obtained by assuming the partial sums already approach a constant and then truncating the very Bessel series whose convergence is in question. The Hölder/Young route in Theorem 2.11 relies on false regularity premises and applies a finite-interval criterion to an improper integral, so it also presupposes the tail convergence. References to Alekseyev and Meiburg are external and legitimate, but they only state that convergence would imply μ(π)≤5/2; they do not supply the convergence that is assumed. No machine-checked or independent numerical bound for the tail is provided. Because the principal 'result' reduces to assuming the convergence it claims to prove, the circularity score is high.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The central numerical claim rests on two fitted constants: delta, tuned to the claimed sum, and c1, taken from a truncated partial sum. The main analytic arguments rely on false Holder and uniform-lower-bound assumptions, and on inserting the target series into its own tail. The elliptic-curve classes are speculative and not load-bearing.

free parameters (3)
  • delta = 0.23 <= delta <= 0.232942
    Introduced as a uniform lower bound on |sin(k)| in Section 2.8, then tuned by Eq (2.38) to match the claimed series value 30.314510. No such uniform positive lower bound exists because pi is irrational.
  • c1 = ≈ 78.1160806386
    Defined as the limit of Lambda(sigma) in Eq (1.6), computed by truncating the Bessel sum at sigma=10000 via WolframAlpha and treated as the exact limit; used in Eqs (1.7) and (2.37) to produce the series value.
  • L = 1/constant, symbolic
    Introduced in Section 1.1 after identifying a constant in the recursive trigonometric substitution; the formula L = 1/constant makes the final expression (1.34) tautological.
assumptions (4)
  • standard math Young's criterion: if f is alpha-Holder, g is beta-Holder, alpha+beta>1 and no common discontinuities, then the Riemann-Stieltjes integral exists.
    Invoked in Theorems 1.2, 2.8, 2.11, and 2.14. The theorem itself is standard, but the hypotheses are not met by the functions used.
  • domain assumption f(x)=csc^2(x)/x^3 is 1-Holder continuous and floor(x) is 1-Holder continuous.
    Asserted in Section 2.2.3 before Theorem 2.11; false, because csc^2 has poles at multiples of pi and floor is not Lipschitz across integer boundaries.
  • domain assumption There exists delta > 0 with |sin(k)| >= delta for all integers k.
    Used in Section 2.8 to bound sum 1/(k sin k)^{4m/(2m-1)} by zeta(2)/delta^2; false because pi irrational implies inf_k |sin(k)| = 0.
  • ad hoc to paper The tail series Theta can be replaced by an expression containing sum_{n=sigma}^infinity csc^2(n)/n^3 without assuming its convergence.
    Occurs in Eqs (2.18)-(2.19) and in the rearrangement (2.22)-(2.24). This is the core circular step.
invented entities (1)
  • Classes C_{a_j,b_j} of integers satisfying csc^2(lambda)/lambda^3 = 1 + a_j/lambda^2 + b_j/lambda^3 on Weierstrass curves
    purpose: To model Flint-Hills partial sums via polygamma functions and elliptic curves in Section 4
    No evidence that arbitrary integers lie on such curves; the coefficients are chosen by regression and the construction is speculative.

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Pith. "Pith review of Diophantine FLINT-HILLS series." pith.science (2026). https://pith.science/paper/OBMNSGZN

@misc{pith2026250203474,
  author       = {Pith},
  title        = {Pith review of: Diophantine FLINT-HILLS series},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBMNSGZN}},
  note         = {Machine review of arXiv:2502.03474}
}
read the original abstract

We prove the convergence of the FLINT-HILLS series and establish new criteria for a similar type of diophantine or lacunary series, which faces issues due to spaced long terms coming from the trigonometric nature of functions, e.g., cosecant in the FLINT-HILLS series. We connect the FLINT-HILLS series to the Fermi-Dirac integral via the Riemann-Stieltjes integral and YOUNG'S inequality criteria but also proved that the upper bound of the irrationality measure of pi is equal or lower than 2.5 expected if the FLINT-HILLS series converged.

Figures

Figures reproduced from arXiv: 2502.03474 by the authors.

Figure 1
Figure 1. Slope field of the first-order ordinary differential equation Λ ′ (t) = − √π 3 ψ ′′′(t), where ψ ′′′(t)is the third derivative of the digamma function with respect to t. Corollary 2.5. Given Λ(t) from Corollary 2.4, its derivatives are defined by Λ (m) (t) = − π √ 3 ψ (m+2)(t), where m ≥ 1, and ψ (m+2)(t) denotes the m-th derivative of the digamma function with respect to t. Proof. As proved via Corollary 2.4, we ca… view at source ↗
Figure 2
Figure 2. This representation aids in understanding how the discon￾tinuous nature of the floor function interacts with the integrand func￾tion f(x). By visually showing the contributions at each integer step, it provides an intuitive grasp of how the Riemann-Stieltjes integral ac￾cumulates values over intervals where ⌊x⌋ changes [PITH_FULL_IMAGE:figures/full_fig_p038_2.png] view at source ↗

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