REVIEW 3 major objections 4 minor 9 references
A Note on a Recent Attempt to Prove the Irrationality of $\zeta(5)$
T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read This note argues that Suman's proof of the irrationality of $\zeta(5)$ fails because the induction's base case has integer solutions $a=2b$ and $a=b$, leaving the irrationality of $\zeta(5)$ unproven.
desk verdict The note's base-case critique of Suman's zeta(5) proof probably conflates an unconditional Diophantine equation with the conditional one derived under ζ(5)=a/b, so the refutation doesn't land. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the linear Diophantine equation $d_n a - 2d_n b = -k_i b$ quoted from Suman's Eq. (47)/(48), together with the induction on $n$ used to assert that it has no integer solutions. The note's argument works by testing the base case $n=1$ explicitly and producing solutions $a=2b$ and $a=b$, which shows that the equation's solvability is independent of the irrationality of $\zeta(5)$. The note also sets out Proposition 3.1, the standard irrationality criterion requiring rational approximations whose common denominator grows slower than the reciprocal approximation error, and observes that Suman's integrals do not satisfy that criterion.
What would settle it
Check Suman's arXiv:2407.07121v6 directly: if Eq. (48) is stated with the condition $n\ge b$ or with $b>1$, or if the induction is shown only for $n$ large enough, then the $n=1$ counterexamples are outside the claimed range. Alternatively, exhibit the full derivation of Eq. (48) and see whether the solutions $a=2b$ and $a=b$ satisfy every stated hypothesis; a second check is to run the same induction at $n=2$ and $n=3$ and see whether integer solutions persist.
Extended reading notes
Core claim
The paper's central claim is that Eq. (48) of Suman's preprint admits integer solutions, contrary to the assertion used to support the irrationality of $\zeta(5)$. Specifically, at the base case $n=1$ the Diophantine equation $a-2b=-k_i b$ with $0\le k_i\le 1$ and $1\mid k_i b$ forces $a=2b$ or $a=b$. Since these are genuine integer solutions, the induction's base case collapses, and with it the proof of Suman's Theorem 1; the same type of conflation is said to appear in Theorem 2 for $\zeta(2m+1)$.
Load-bearing premise
The diagnosis depends on the quoted equations and conditions from Suman's preprint being accurate and complete; if Suman's original argument imposes additional restrictions, such as $b>1$, $n\ge b$, or $a,b$ coprime, that eliminate the $n=1$ solutions, the claimed base-case flaw would not apply as stated.
Editorial extensions
If this is right
- If the diagnosis is right, Suman's Theorem 1 does not establish that $\zeta(5)$ is irrational.
- The same fallacy is claimed for Theorem 2, so the purported irrationality of $\zeta(2m+1)$ for $m\ge 2$ is likewise not established.
- Known results remain intact: $\zeta(3)$ is irrational, at least one of $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational, and the odd zeta values span an infinite-dimensional space.
- The standard irrationality criterion of Proposition 3.1 is not met by the integrals in Suman's Lemma 1, according to the note's numerical check.
Reading between the lines
- A fuller refutation would need the complete text of Suman's induction; if the original imposes $n\ge b$ with $b>1$ or a coprimality condition, the $n=1$ solutions might lie outside the stated range, so the base-case blow-up is not yet clinched by the quoted excerpt alone.
- One way the "absurdity" move could be legitimate is if Eq. (48) had been derived under the explicit assumption $\zeta(5)=a/b$ with $a,b\in\mathbb{Z}$; the note says Suman invokes $\zeta(5)\notin\mathbb{Z}$ after the fact, but checking the original derivation would settle this.
- A testable extension is to analyze the same induction at small $n>1$; if integer solutions persist at every $n$, the flaw is systematic, and if not, the failure is confined to the base case but still fatal.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The note argues that Shekhar Suman's claimed proof of the irrationality of zeta(5) (arXiv:2407.07121v6) is invalid. The central allegation is that the induction proof of the key Diophantine claim fails at the base case n=1, because the equation admits integer solutions a=2b and a=b, and that Suman's dismissal of these as 'absurd' conflates the algebraic solvability of the equation with the arithmetic nature of zeta(5). The note also asserts, without numerical detail, that Suman's integrals do not satisfy the growth condition of a standard irrationality criterion, and it closes with a survey of recent work by Calegari, Dimitrov, and Tang.
Significance. If the critique were established, it would show that a claimed proof of zeta(5)'s irrationality is flawed and that the irrationality of zeta(5) remains open. The paper usefully directs attention to the precise logical role of Diophantine equations in irrationality proofs, and the presentation of the standard criterion in Proposition 3.1 is clear. However, the central objection depends on the exact quantifier structure and index ranges of Suman's induction, which the note does not reproduce, and the numerical claim is unsupported; the paper's current value is therefore conditional.
major comments (3)
- [§2, displayed equations surrounding Eq. (48)] The note's formulation of the induction claim changes the index range from "1 <= k_i <= d_n - 1, n >= b" (quoted from [6, Eq. (47)]) to "0 <= k_i <= d_n, n >= 1". This is materially important: since the text itself notes that 1 | k_i b at n=1, taking d_1=1 gives no admissible k_i in the original range, whereas the modified range admits k=0 and k=1 and yields a=2b and a=b. To show that the base case actually invalidates Suman's induction, the note must reproduce Suman's exact statement and verify d_1, the index range, and the lower bound n >= b.
- [§2, discussion of Eq. (48) and the 'absurd' solutions] The central objection conflates an unconditional Diophantine claim with the conditional statement arising under the hypothesis zeta(5)=a/b. If a and b are introduced as numerator and denominator of zeta(5), then the solutions a=2b and a=b imply zeta(5)=2 and zeta(5)=1; since 1<zeta(5)<2, these are contradictions and can legitimately discharge the base case. The note's assertion that solvability is "logically independent" of whether zeta(5) is an integer holds only if the equation is considered apart from the rationality hypothesis, but the note does not show that Suman's induction claim has that unconditional form. Quoting the induction hypothesis and the derivation of Eqs. (47)-(48) is necessary to establish the purported logical flaw.
- [§3, remark after Proposition 3.1] The claim that "Numerical verification shows that the I_n in [6, Lemma 1] does not satisfy the condition (3) of Proposition 3.1" is unsupported: no numerical values, no table, and no description of the computation are provided. In addition, the remark presupposes that Suman's proof is an instance of Proposition 3.1; the note does not establish that Suman invoked this criterion. The claim should be substantiated or removed, since as it stands it cannot support the conclusion that Suman's method cannot prove the irrationality of zeta(5).
minor comments (4)
- [§2, penultimate paragraph] The assertion that "similar logical fallacies" occur in the proof of [6, Theorem 2] is made without any analysis of that theorem; either provide details or soften the claim.
- [§3, opening sentence] The phrase "show that Suman's method does not meet the criterion" is imprecise: Proposition 3.1 is one sufficient criterion, and failure to meet it is not by itself a flaw in a proof that does not invoke it.
- [Abstract and §3] There are minor typographical issues: "dis cuss" in the abstract should be "discuss", and "the In" in §3 should be "the I_n".
- [§4] The survey of Calegari-Dimitrov-Tang is not connected to the preceding critique; relating it to the question of admissible integral representations for zeta(5) would improve the focus.
Circularity Check
No significant circularity: the critique of Suman's base case is grounded in direct substitution and external standards, not in the claim being critiqued.
full rationale
This note is a critical analysis of an external proof, not a derivation that predicts or explains its own inputs. The central claim—that Suman's induction base case admits integer solutions—is established by direct substitution into the quoted equation (for n=1, a−2b=0 or a−2b=−b), not by assuming ζ(5) irrationality. The note's mention that ζ(5) is known not to be an integer is used only to describe and criticize Suman's own 'absurd' dismissal, not as a load-bearing premise for the critique. No parameters are fitted to data, no prediction is derived from fitted inputs, and no load-bearing result is imported from the authors' own prior work. All citations are to external prior work (Apéry, Beukers, Ball-Rivoal, Zudilin, Calegari-Dimitrov-Tang) and are used for context, standard criteria, or background, not to force the conclusion. Even if the mathematical objection to Suman's proof were debatable due to the quantifier structure of the induction hypothesis, that would be a correctness concern, not circularity. The paper is therefore self-contained with respect to its own argument and exhibits no circular step.
Assumptions & free parameters
assumptions (1)
- domain assumption The equations and statements quoted from Suman's arXiv preprint accurately reflect the original argument.
Cite this review
Pith. "Pith review of A Note on a Recent Attempt to Prove the Irrationality of $\zeta(5)$." pith.science (2026). https://pith.science/paper/UQCHWXHB
@misc{pith2026241116774,
author = {Pith},
title = {Pith review of: A Note on a Recent Attempt to Prove the Irrationality of $\zeta(5)$},
year = {2026},
howpublished = {\url{https://pith.science/paper/UQCHWXHB}},
note = {Machine review of arXiv:2411.16774}
}
abstract
Recently Shekhar Suman [arXiv: 2407.07121v6 [math.GM] 3 Aug 2024] made an attempt to prove the irrationality of $\zeta(5)$. But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.
Reference graph
Works this paper leans on
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[1]
Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque, 61 (1979) 11–13
R. Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque, 61 (1979) 11–13
work page 1979
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[2]
K. Ball and T. Rivoal, Irrationalité d’une infinité de val eurs de la fonction zêta aux entiers impairs, Invent. Math. , 146.1 (2001) 193–207
work page 2001
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[3]
Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull
F. Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. Lond. Math. Soc. , 11.3 (1979) 268–272
work page 1979
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[4]
F. Calegari, V. Dimitrov, and Y. Tang, The linear indepen dence of 1, ζ(2), and L(2, χ − 3), arXiv preprint, arXiv:2408.15403, 2024
arXiv 2024
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[5]
D. Huylebrouck, Similarities in Irrationality Proofs f or π , ln 2, ζ(2), and ζ(3), The American Mathemat- ical Monthly, 108.3 (2001) 222–231
work page 2001
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[6]
A note on the Irrationality of $\zeta(5)$ and Higher Odd Zeta Values
S. Suman, A note on the irrationality of ζ(5) and higher odd zeta values, arXiv preprint , arXiv:2407.07121v6, 2024
work page Pith review arXiv 2024
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[7]
M. Waldschmidt, Diophantine approximation on linear algebraic groups: tra nscendence properties of the exponential function in several variables , Vol. 326, Springer Science & Business Media, 2013
work page 2013
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[8]
Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational, Uspekhi Mat
W. Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational, Uspekhi Mat. Nauk , 56.4 (2001) 149–150
work page 2001
Show all 9 references
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[9]
Zudilin, Analytic Methods in Number Theory: When Complex Numbers Count , Vol
W. Zudilin, Analytic Methods in Number Theory: When Complex Numbers Count , Vol. 11, World Scientific, 2023. 5
2023
Reviewed August 12, 2026 · model on record in the stance chip above.
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