REVIEW 3 major objections 4 minor 8 references
On the vanishing of coefficients of $\eta^{26}
T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read The paper proves a partial converse to the earlier vanishing criterion for coefficients of $\eta^{26}$, giving an if-and-only-if on two arithmetic progressions.
desk verdict A promising partial converse to Serre's vanishing criterion for eta^26, but the proof as printed has a load-bearing gap in the nonvanishing of odd-power CM coefficients. 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 load-bearing machinery is the decomposition of $\eta^{26}$ into four CM-type Hecke eigenforms, meaning eigenforms whose coefficients come from Hecke characters of imaginary quadratic fields, as obtained in [7]. After the change of variable $z\mapsto 12z$, one obtains the exact identity $$p_{26}(n)=\frac{1}{32617728}\bigl(t_{1+}(12n+13)+t_{1-}(12n+13)-t_{2+}(12n+13)-t_{2-}(12n+13)\bigr),$$ where $t_{1\pm}$ come from $\mathbb{Q}(\sqrt{-3})$ and $t_{2\pm}$ from $\mathbb{Q}(i)$. Multiplicativity of eigenform coefficients splits each $t(12n+13)$ into a product over prime powers $p^{\alpha}$, and the Hecke recurrences (3)--(8) reduce each $t(p^{\alpha})$ to powers of $t(p)$. The paper adds two tools: congruence analyses modulo $5$ and $7$ (Propositions 3.5--3.7) that control divisibility of $t_{2+}(p^{\alpha})$ and $t_{1+}(p^{\alpha})$ for $p\equiv 1\pmod{12}$, and 2-adic nonvanishing lemmas (3.8--3.11) showing $t_{1+}(p^{\alpha})-t_{2+}(p^{\alpha})\neq 0$ for even $\alpha>1$ and primes $p\equiv 1,5,7\pmod{12}$. Together these force the four components not to cancel in the restricted progressions.
What would settle it
Take a small $n$ satisfying the hypotheses of Theorem 1.4 in which a $3\bmod 4$ prime and a $2\bmod 3$ prime both divide $12n+1$ to odd order and compute $p_{26}(25n+1)$ directly from the product definition of $\eta^{26}$: if the coefficient is nonzero, the sufficient direction fails. Take instead an $n$ satisfying Theorem 4.6's even-exponent condition and compute $p_{26}(25n+1)$: a zero value would refute the nonvanishing lemma. Scanning the first few thousand such $n$, using the identity above, would settle both directions.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is that the earlier two-prime condition is not just sufficient but also necessary inside these families. For $n$ satisfying $\mathrm{ord}_p(12n+1)\not\equiv 4\pmod 5$ for every prime $p\equiv 1\pmod{12}$, the paper proves $p_{26}(25n+1)=0$ if and only if some $p\equiv 3\pmod 4$ and some $p'\equiv 2\pmod 3$ both divide $12n+1$ to odd order (Theorem 1.4); replacing $25$ by $49$, $5$ by $7$, and the arithmetic progression by $49n+3$ gives the analogous Theorem 1.5. The converse is proved contrapositively: if all $3\bmod 4$ exponents are even or all $2\bmod 3$ exponents are even, then the coefficient cannot vanish. The proof writes $\eta^{26}$ as a sum of four CM-type Hecke eigenform components and uses divisibility congruences modulo $5$ and $7$, together with 2-adic nonvanishing lemmas, to show that the components cannot cancel.
Load-bearing premise
The proof stands on the exact identity expressing each $p_{26}(n)$ as a fixed rational multiple of the four auxiliary sequences; if a sign or the normalizing constant $32617728$ in that formula were wrong, all later divisibility arguments would describe the wrong function.
Editorial extensions
If this is right
- For every $n$ in the Theorem 1.4 family, the zeros of $p_{26}(25n+1)$ coincide exactly with the odd-order presence of both a $3\bmod 4$ and a $2\bmod 3$ prime in $12n+1$.
- The same exact description holds for $p_{26}(49n+3)$ under the Theorem 1.5 restriction.
- Theorems 4.1--4.5 supply explicit infinite families where $p_{26}(n)\neq 0$, including $12n+13$ a prime power whose prime is not $11\bmod 12$.
- The 2-adic lemmas show $t_{1+}(p^{\alpha})-t_{2+}(p^{\alpha})\neq 0$ for every even $\alpha>1$ in the relevant residue classes, so any zero in the restricted families must come from the odd-exponent prime factors, matching the earlier sufficient condition.
- Within the Theorem 1.4 family, the excluded class $\mathrm{ord}_p(12n+1)\equiv 4\pmod 5$ is exactly where the mod-5 divisibility argument would break down, so the stated restriction is a sharp boundary for the method.
Reading between the lines
- A natural extension, not claimed in the paper: replacing the moduli $5$ and $7$ by other primes and using multipliers such as $\ell^2(12n+1)$ could lift the congruence restrictions, because those restrictions only exclude exponent classes where one of the Hecke coefficients is divisible by the modulus.
- If the full converse eventually holds, then the zeros of $p_{26}$ would split into two clean arithmetic profiles: odd exponents in both complementary residue classes, or square totals made only of $11\bmod 12$ primes, placing $\eta^{26}$ on the same footing as the lower even powers with known necessary-and-sufficient conditions.
- One testable computational consequence of the divisibility lemmas is that within the Theorem 1.4 family, $t_{1+}(12n+13)-t_{2+}(12n+13)$ should never be divisible by $5$; checking this independently of $p_{26}$ would probe the congruence core of the proof.
- The proof's 2-adic argument could be pushed further to produce an explicit lower bound on the 2-adic valuation of $t_{1+}(p^{\alpha})-t_{2+}(p^{\alpha})$, giving quantitative control on how large $p_{26}$ must be when it is nonzero in these families.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the integer coefficients p_26(n) of eta^26. Serre gave a sufficient condition for p_26(n)=0 in terms of the parity of the exponents in 12n+13 for primes congruent to 3 mod 4 and 2 mod 3, plus a square case. The authors prove partial converses: for integers n with ord_p(12n+1) not congruent to 4 mod 5 (resp. 6 mod 7) for every prime p congruent to 1 mod 12, they claim that p_26(25n+1)=0 (resp. p_26(49n+3)=0) iff 12n+1 has an odd exponent at some prime p=3 mod 4 and an odd exponent at some prime p'=2 mod 3. The proof uses Serre's decomposition of eta^26(12z) into four CM eigenforms, the Hecke multiplicativity relations (2), the recurrences (3)-(8), and congruences modulo 5 and 7.
Significance. If the main theorems were correct, they would be a genuine partial converse to Serre's theorem in an infinite family, and the method is a natural one: reducing the nonvanishing question to local CM coefficients and controlling them by congruences. The paper is transparent about using Serre's sufficiency theorem as external input, and it does not fit parameters. The explicit formulas for t_1± and t_2± are checkable, and several of the congruence propositions are stated with enough detail to be verified. However, the two gaps described below are load-bearing for the claimed converses, so the significance is real but conditional.
major comments (3)
- [Theorem 4.2 and Theorem 4.5] In Theorem 4.2 the displayed factorization of t_2+(m)+t_2-(m) is prod_{p≠5} t_2+(p^α_p) times (1+(-1)^μ) times prod_{even} t_2+(p^α_p) times prod_{odd} t_2+(p^α_p), and Theorem 4.5 has the analogous factorization for t_1+(m)+t_1-(m). Proposition 3.5(i) shows 5 ∤ t_2+(p^{2α}) for p ≡ 5 mod 12, and Proposition 3.6(i) shows 5 ∤ t_1+(p^{2α}) for p ≡ 7 mod 12, but neither result gives nonvanishing of the odd-exponent factors that occur when μ>0 or ν>0. Since t_1±(m) is forced to vanish in Theorem 4.2, and t_2±(m) is forced to vanish in Theorem 4.5, a single zero among those odd-exponent factors would make p_26(m)=0, contradicting the claimed theorem. Moreover, the factor over p≠5 in Theorem 4.2 includes primes p ≡ 1 mod 12 for which the paper proves no nonvanishing of t_2+(p^α); the analogous point applies to the factor over p≠7 in Theorem 4.5. The proofs of Theorems 4.6 and 4.7 use exactly Theorems 4.2 and 4.5 in the branch with an odd exponent of a 5 mod 12 or 7 mod 12 prime, so Theorems 1.4 and 1.5 are not established.
- [Lemmas 3.10 and 3.11] In Lemmas 3.10 and 3.11 the substitution of t_2+(p)^2 = 2c (resp. t_1+(p)^2 = 2d) is algebraically inconsistent. From t_2+(p)^2 = 2c it follows that t_2+(p)^{2β} = (2c)^β = 2^β c^β and t_2+(p)^{2β-2} = (2c)^{β-1} = 2^{β-1} c^{β-1}, but the displayed expressions in Lemma 3.10 replace these powers with 2^{2β} c^{2β} and 2^{2β-2} c^{2β-2}; Lemma 3.11 does the same with d. The 2-adic valuation argument in these lemmas therefore does not prove the stated nonvanishing of t_1+(p^α)-t_2+(p^α) for p ≡ 5 or 7 mod 12 and even α > 1. Since Theorem 4.1 relies on these lemmas for p ≡ 5 and 7 mod 12, Theorem 4.1 is not proved as printed.
- [Theorem 4.1] The last sentence of the proof of Theorem 4.1 states that 'α is odd when p ≡ 5 or 7 (mod 12)'. This is false: 12n+13 ≡ 1 (mod 12), so for p ≡ 5 or 7 (mod 12) the exponent α must be even. As written this contradicts the appeal to Lemmas 3.10 and 3.11, which are stated for even α > 1. The parity should be corrected to 'even'; the theorem may be repairable, but the current text is internally inconsistent.
minor comments (4)
- [Lemma 3.8] In Lemma 3.8 the roles of t_1+(p) and t_2+(p) are swapped relative to the conventions in (14) and (15): the x+iy expression is assigned to t_1+(p) and the z+w√-3 expression to t_2+(p), although according to (14)-(15) it should be the other way around. The conclusion is unaffected because the proof uses the difference, but the notation should be aligned.
- [Proposition 3.6(ii)] The expression '7 ∤ t_1+(7)√-3' should read '7 ∤ t_1+(7)/√-3' (and similarly for the surrounding divisibility statements); the current notation is ambiguous.
- [Proposition 3.7(iv)] In the proof of Proposition 3.7(iv), the line 't1+ ≡ 2 + 4.3(xy)2 + 3 ≡ ±2 (mod 5)' is unclear; the '4.3' should presumably be '4·3', and the display should be rewritten.
- [Throughout] There are minor typos: 'The authors would like thank' should be 'would like to thank', and the affiliation line misspells the first author's name as 'Krishnamoorhty'.
Circularity Check
No circularity: the converse theorems are derived from Serre's external eigenform decomposition and independent Hecke-coefficient estimates, not from the result being proved.
full rationale
The paper's central results (Theorems 1.4 and 1.5) are genuine partial converses to Serre's sufficient condition. The forward direction is imported from Serre's Theorem 1.1, while the converse direction is proved separately through the Hecke eigenform decomposition in equation (1), the multiplicativity relation (2), and the coefficient estimates in Propositions 3.5-3.7 and Lemmas 3.8-3.11. No parameter is fitted, no data subset is used to force a conclusion, and no load-bearing premise is justified only by a self-citation: the reference list contains no prior work of Krishnamoorthy or Dalal, and the cited external results by Serre [7] and Chan-Cooper-Toh [1] are independent published theorems. The final implications p26(25n+1) ≠ 0 and p26(49n+3) ≠ 0 are obtained by exhibiting prime divisibility or 2-adic nonvanishing of the constituent CM eigenform coefficients, which is not equivalent by construction to the condition being tested. For completeness, the skeptical gap in Theorems 4.2 and 4.5 is a correctness issue, not a circularity issue: after the factorization t2+(m)+t2-(m)=(1+(-1)^mu) times a product of t2+(p^{alpha_p}) (and analogously for t1+ in Theorem 4.5), the proof does not supply a separate lemma proving the relevant product is nonzero for odd alpha_p. A missing nonzero lemma could invalidate the converse proof, but it does not make the derivation reduce to its own input, so it does not raise the circularity score.
Assumptions & free parameters
assumptions (5)
- domain assumption Identity p26(n)=1/32617728(t1+(12n+13)+t1-(12n+13)-t2+(12n+13)-t2-(12n+13)) from Serre [7]
- domain assumption The functions c1± and c2± defined in Section 2 are Hecke characters of exponent 12 with the stated conductors and induce the stated Dirichlet character
- standard math Coefficient recurrences (3)-(8) for primes p mod 4
- domain assumption Finite-field polynomial range statements in Propositions 3.5(ii) and 3.7(ii)
- domain assumption Sufficiency direction of Theorems 1.4 and 1.5 relies on Serre's Theorem 1.1
Cite this review
Pith. "Pith review of On the vanishing of coefficients of $\eta^{26}." pith.science (2026). https://pith.science/paper/PY4X2OL5
@misc{pith2026241114990,
author = {Pith},
title = {Pith review of: On the vanishing of coefficients of $\eta^26},
year = {2026},
howpublished = {\url{https://pith.science/paper/PY4X2OL5}},
note = {Machine review of arXiv:2411.14990}
}
abstract
J.-P. Serre, in his paper [1], established a sufficient condition on $n$ for the $n$-th coefficient of the series $\eta^{26}$ to vanish. However, the question that whether this is a necessary condition remained unanswered. In this paper, using the theory of Hecke eigenforms explored by Serre, we prove some partial cases for the converse part.
Reference graph
Works this paper leans on
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[1]
H. H. Chan, S. Cooper, and P. C. Toh, The 26th power of Dedek ind’sη-function, Adv. Math. 207(2) (2006), 532–543
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[6]
R. A. Rankin, Hecke operators on congruence subgroups of the modular group, Math. Ann. 168 (1967), 40–58
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[7]
Serre, Sur la lacunarit´ e des puissances de η, Glasgow Math
J.-P. Serre, Sur la lacunarit´ e des puissances de η, Glasgow Math. J. 27 (1985), 203–221
work page 1985
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[8]
http://magma.maths.usyd.edu.au/calc/ S. Krishnamoorhty, Indian Institute of Science Education a nd Research, Thiruvananthapuram, India Email address : srilakshmi@iisertvm.ac.in Tarun Dalal, Institute of Mathematical Sciences, Shanghai Tech University 393 Middle Huaxia Road, Pudong, Shanghai 201210, China Email address : tarun.dalal80@gmail.com
Reviewed August 12, 2026 · model on record in the stance chip above.
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