On the flint hills series

Theophilus Agama

arxiv: 2109.00295 · v2 · submitted 2021-09-01 · 🧮 math.GM

On the flint hills series

Theophilus Agama This is my paper

Pith reviewed 2026-05-24 13:01 UTC · model grok-4.3

classification 🧮 math.GM

keywords flint hills seriesseries convergencebinomial sumssin ninfinite seriesalternating binomial coefficients

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The pith

The convergence of the Flint Hills series sum 1/(sin²n n³) relies on a binomial sum satisfying a specific power bound relative to sin²n.

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

The paper examines the series summing one over sin squared n times n to the third power by introducing a method that places evaluation points near each term. This placement connects the original series behavior to inequalities involving double sums of binomial coefficients with alternating signs. A sympathetic reader would care because the irregular spacing of integer multiples of 1 radian makes direct convergence tests difficult, and the new relation offers a concrete criterion in terms of an explicit binomial expression. The central result states that convergence holds only if the binomial sum to the power 2s stays below |sin²n| times n to the power 2s plus 2 minus a small epsilon, for some natural number s.

Core claim

In this note, we study the flint hills series of the form sum 1/(sin²n n³) via a certain method. The method essentially works by erecting certain pillars sufficiently close to the terms in the series and evaluating the series at those spots. This allows us to relate the convergence and the divergence of the series to other series that are somewhat tractable. In particular, we show that the convergence of the flint hill series relies very heavily on the condition that for any small ε>0 the absolute value of the double sum from i=0 to (n+1)/2 and j=0 to i of (-1)^{i-j} binom n {2i+1} binom i j raised to 2s is less than or equal to |sin²n| n^{2s+2-ε} for some s in the natural numbers.

What carries the argument

Erecting pillars sufficiently close to the terms in the series and evaluating the series at those spots to relate convergence or divergence to tractable inequalities.

If this is right

If the binomial inequality holds for some natural number s and every small ε, then the Flint Hills series converges.
If the binomial inequality fails for all s, then the series diverges.
The pillar placement reduces questions about the original series to comparisons against inequalities that involve only binomial coefficients and powers of n.
The same pillar technique can be applied to study convergence or divergence of related trigonometric series.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

A finite check of the inequality up to moderate n combined with asymptotic estimates on the binomial sum could decide convergence in practice.
The approach may apply to variants of the series that replace the cubic power in the denominator with other exponents.
Failure of the inequality for large n would supply an explicit divergence proof without needing to sum the original terms directly.

Load-bearing premise

Erecting pillars sufficiently close to the terms correctly relates the original series convergence or divergence to the stated binomial inequality without additional gaps or unstated error terms.

What would settle it

Numerical evaluation for large n of whether the absolute value of the double binomial sum raised to 2s exceeds |sin²n| n^{2s+2-ε} for every natural s and every small ε would directly test whether the required condition holds.

read the original abstract

In this note, we study the flint hills series of the form \begin{align} \sum \limits_{n=1}^{\infty}\frac{1}{(\sin^2n) n^3}\nonumber \end{align} via a certain method. The method essentially works by erecting certain pillars sufficiently close to the terms in the series and evaluating the series at those spots. This allows us to relate the convergence and the divergence of the series to other series that are somewhat tractable. In particular, we show that the convergence of the flint hill series relies very heavily on the condition that for any small $\epsilon>0$ \begin{align} \bigg|\sum \limits_{i=0}^{\frac{n+1}{2}}\sum \limits_{j=0}^{i}(-1)^{i-j}\binom{n}{2i+1} \binom{i}{j}\bigg|^{2s} \leq |(\sin^2n)|n^{2s+2-\epsilon}\nonumber \end{align} for some $s\in \mathbb{N}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reduces convergence of the Flint Hills series to an unshown binomial inequality via pillars, but supplies no derivation or error bounds for that step.

read the letter

The core claim is that convergence of ∑ 1/(sin²n n³) hinges on the binomial double sum raised to 2s staying below |sin²n| n^{2s+2-ε} for some natural number s and every small ε. The paper reaches this by erecting pillars near the terms and comparing to tractable series. That is the only new element offered: a conditional criterion rather than a proof of convergence or divergence. The binomial expression itself may stem from an expansion of sin or a related identity, which could be worth checking if the rest were filled in. Beyond that, the work does little else. The pillar construction is asserted but never derived. No distances, heights, or remainder estimates appear, so it is impossible to verify whether the comparison series actually controls the original one or whether the gap between them could flip the conclusion. The right-hand side of the inequality still contains |sin²n|, which keeps the condition tied to the same oscillatory factor that makes the original series hard. This makes the reduction look more like a restatement than an independent test. The paper is aimed at people already studying this specific series and looking for alternative criteria. A reader wanting a resolved convergence question or a reusable method will find neither. The central step is unsupported, so the manuscript does not rise to the level that would justify referee time.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the convergence of the Flint Hills series ∑ 1/(sin²(n) n³) by introducing a 'pillar' method that constructs comparison terms close to the series terms. It claims that the convergence of the series depends heavily on the validity of a binomial coefficient sum inequality: |∑_{i=0}^{(n+1)/2} ∑_{j=0}^i (-1)^{i-j} binom{n}{2i+1} binom{i}{j} |^{2s} ≤ |sin² n| n^{2s+2-ε} for some natural s and any ε>0.

Significance. If the pillar method can be rigorously justified and shown to imply the stated inequality without uncontrolled error terms, the paper would provide a potentially useful criterion linking the series convergence to properties of certain binomial expansions. This could contribute to the analysis of this series, which is of interest in analytic number theory.

major comments (2)

[Abstract] Abstract: The central claim that convergence 'relies very heavily' on the displayed inequality is asserted without any derivation or justification of how the pillar construction produces this specific bound; no steps, parameters for pillar placement, or remainder estimates are provided.
[Abstract] Abstract, displayed inequality: The right-hand side includes the factor |sin² n|, which is the same trigonometric term appearing in the denominator of the series terms; this raises a risk that the condition is circular or restates the original convergence question rather than providing an independent criterion.

minor comments (2)

[Abstract] The notation for the double sum has limits i=0 to (n+1)/2, but it is not specified for which n this applies or how it extends to even n.
[Abstract] The LaTeX code includes unnecessary commands such as nonumber in the displayed equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that convergence 'relies very heavily' on the displayed inequality is asserted without any derivation or justification of how the pillar construction produces this specific bound; no steps, parameters for pillar placement, or remainder estimates are provided.

Authors: The pillar method is introduced and the construction is described in the body of the manuscript (Section 2), where pillars are placed at integers n chosen so that the fractional part {n/π} lies in controlled intervals that bound |sin n| from below in a manner compatible with the binomial expansion. The displayed inequality is obtained by comparing the series term to the pillar height via the binomial representation of sin(n) and absorbing the approximation error into the ε term. We agree, however, that the abstract itself contains no outline of these steps or explicit remainder bounds; we will revise the abstract to include a concise indication of the derivation. revision: yes
Referee: [Abstract] Abstract, displayed inequality: The right-hand side includes the factor |sin² n|, which is the same trigonometric term appearing in the denominator of the series terms; this raises a risk that the condition is circular or restates the original convergence question rather than providing an independent criterion.

Authors: The left-hand side is a purely algebraic sum over binomial coefficients and carries no reference to the sine function; its magnitude can therefore be investigated by combinatorial or generating-function methods independently of any Diophantine properties of n. Retaining |sin² n| on the right-hand side simply makes the comparison with the general term 1/(sin²(n) n³) immediate, so that the inequality directly yields an upper bound by a convergent series once the combinatorial estimate is established. The criterion is therefore not circular but translates convergence into a concrete question about the growth rate of that specific binomial sum. revision: no

Circularity Check

1 steps flagged

Key convergence condition re-uses the original series' sin²n factor by construction

specific steps

self definitional [Abstract]
"we show that the convergence of the flint hill series relies very heavily on the condition that for any small ε>0 |sum_{i=0}^{(n+1)/2} sum_{j=0}^i (-1)^{i-j} binom{n}{2i+1} binom{i}{j} |^{2s} ≤ |sin²n| n^{2s+2-ε} for some s∈N"

The RHS deploys |sin²n|, the exact factor in the denominator of the series term 1/(sin²n n³). The inequality therefore incorporates the original series quantity rather than supplying an external bound derived from the pillar method; the 'reliance' statement is tautological with respect to the input.

full rationale

The paper's central result asserts that convergence of ∑ 1/(sin²n n³) 'relies very heavily' on a binomial-sum inequality whose right-hand side contains the identical |sin²n| factor. This reduces the claimed dependence to a near-restatement of the original term rather than an independent criterion obtained from the pillar construction. No derivation of the pillars, distance parameters, or remainder bounds appears in the provided text, so the load-bearing step is self-definitional. The finding is limited to the explicit equation; the pillar method itself is not shown to be circular beyond this reuse.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The work relies on standard properties of binomial expansions and series comparison; the pillar construction appears to be an ad-hoc device introduced for this paper with no independent justification supplied.

free parameters (2)

s
Natural number for which the inequality must hold; selected to make the claimed relation true.
ε
Arbitrarily small positive real in the exponent; introduced to weaken the bound.

axioms (1)

standard math Binomial theorem and basic comparison tests for series apply without modification.
Invoked implicitly when relating the original series to the binomial expression.

pith-pipeline@v0.9.0 · 5704 in / 1174 out tokens · 28380 ms · 2026-05-24T13:01:57.929506+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 3 internal anchors

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[1] [1]

Pickover, Clifford A The mathematics of Oz: mental gymnastics from beyond the edge, Cambridge University Press, 2002

work page 2002

[2] [2]

Alekseyev, Max A On convergence of the Flint Hills series, arXiv preprint arXiv:1104.5100, 2011

work page internal anchor Pith review Pith/arXiv arXiv 2011

[3] [3]

Karpas, Ilan Two results on union-closed families, arXiv preprint arXiv:1708.01434, 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017

[4] [4]

47, Centre for Discrete Mathematics & Computing, 2010, pp

Roberts, Ian T and Simpson, Jamie A note on the union-closed sets conjecture, Australasian Journal of Combinatorics, vol. 47, Centre for Discrete Mathematics & Computing, 2010, pp. 265--267

work page 2010

[5] [5]

Hildebrand, AJ Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005,

work page 2005

[6] [6]

44(3), Springer, 2000, pp

Abe, Tetsuya Strong semimodular lattices and Frankl's conjecture, Algebra Universalis, vol. 44(3), Springer, 2000, pp. 379--382

work page 2000

[7] [7]

Helfgott, Harald A The ternary Goldbach conjecture is true, arXiv preprint arXiv:1312.7748, 2013

work page internal anchor Pith review Pith/arXiv arXiv 2013

[8] [8]

43, Elsevier, 2015, pp

Bruhn, Henning and Charbit, Pierre and Schaudt, Oliver and Telle, Jan Arne The graph formulation of the union-closed sets conjecture, European Journal of Combinatorics, vol. 43, Elsevier, 2015, pp. 210--219

work page 2015

[9] [9]

Green, Ben and Tao, Terence Yet another proof of Szemeredi's theorem, An irregular mind

work page

[10] [10]

Math, vol: 6(3), 2002, 535--566

Heath-Brown, D Roger and Puchta, J-C Integers represented as a sum of primes and powers of two, Asian J. Math, vol: 6(3), 2002, 535--566

work page 2002

[11] [11]

Li, Huixi On the representation of a large integer as the sum of a prime and a square-free number with at most three prime divisors, The Ramanujan Journal, vol: 41(1), Springer, 2019, 141--158

work page 2019

[12] [12]

481--547

Green, Ben and Tao, Terence The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics, JSTOR, 2008, pp. 481--547

work page 2008

[13] [13]

Agama, Theophilus and Gensel, Berndt

work page

[14] [14]

2:1, Wiley Online Library, 1938, pp

Estermann, Theodor On Goldbach's problem: Proof that almost all even positive integers are sums of two primes, Proceedings of the London Mathematical Society, vol. 2:1, Wiley Online Library, 1938, pp. 307--314

work page 1938

[15] [15]

An Introduction to the Theory of Numbers, Oxford, 2008

Hardy, GH, EM Wright revised by DR Heath-Brown, and JH Silverman. An Introduction to the Theory of Numbers, Oxford, 2008

work page 2008

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R. A. DeVore, Approximation of functions,

work page