{"id":"e75e777d-a948-4501-bbb5-d1a4462f7165","arxiv_id":"2411.14990","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Partial converse to Serre's vanishing criterion for η^26 coefficients: under restrictions on ord_p(12n+1), p26(25n+1)=0 exactly when two specific prime-divisibility conditions hold.","lead":"This paper studies when the coefficients of the 26th power of Dedekind's eta function vanish. It proves partial converses to a theorem of Serre, giving necessary and sufficient conditions for vanishing in families where prime exponents obey congruence restrictions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.2/4.5 rely on an unproved nonvanishing of the product of t2+(p^α) (resp. t1+(p^α)) over odd exponents; if any such coefficient is zero, the converse fails.","rationale":"The reader already flagged the unstated nonvanishing assumption in Theorem 4.2, though the paper's weakest_assumption field points at equation (1). I agree that equation (1) is the bridge to the CM forms, but it is standard and can be checked directly; the more dangerous point is the unproved nonvanishing inside the final nonvanishing arguments, because it affects Theorems 4.6/4.7 and therefore the iff statements. The other defects the reader notes (false parity comment in Theorem 4.1; algebraic inconsistencies in Lemmas 3.10/3.11) are real but peripheral: Theorem 4.1 is not used in the proofs of 1.4/1.5, and Lemmas 3.10/3.11 are not needed for those proofs either. My check would settle whether the concern lands; if the odd-power coefficients are never zero, the proof can be repaired by adding a short lemma. Thus the verdict stays CONDITIONAL, unchanged from the reader.","tokens_in":17846,"tokens_out":17224,"duration_ms":144399,"concrete_test":"Compute t2+(p) and t2+(p^3) (and, if needed, t2+(p^5)) for all primes p≡5 mod12, p≤1000, using the recurrence (3) and the explicit value from (13); if any is zero, Theorem 4.2 fails. Independently, for all m=25(12n+1)≤10^6 satisfying the hypotheses of Theorem 4.6(1), evaluate the product ∏_{p≡5} t2+(p^{α_p}) and compare with p26(m) from (1); a zero product with nonzero p26 would refute the proof, while a zero product with p26=0 would refute the claim. Repeat the same test for t1+(p^{α}) with p≡7 mod12 for Theorem 4.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central converse Theorems 1.4/1.5 are reduced to Theorems 4.6/4.7, whose first cases invoke Theorem 4.2 (when some p≡5 mod12 has odd exponent in 25(12n+1)) and Theorem 4.5 (similarly for p≡7 mod12). In Theorem 4.2, after setting t1±(m)=0, the proof factors t2+(m)+t2-(m) into a parity factor (1+(-1)^μ) and a product of the coefficients t2+(p^{α_p}) for p≡5 mod12. The parity factor is 2, but the coefficient product is never shown to be nonzero. Proposition 3.5(i) establishes 5∤t2+(p^{2α}) for p≡5, so even powers are safe, but the odd powers that occur when μ>0 are not covered by any nonvanishing result; if one of them vanishes, then t2+(m)+t2-(m)=0 and p26(m)=0, breaking the converse. Theorem 4.5 has the identical gap for t1+(p^{α}) with p≡7 mod12 and α odd, since Proposition 3.6 only addresses even powers. This is a missing step in the exact implication used to prove p26≠0, so the paper as printed does not rigorously establish Theorems 1.4/1.5.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18144,"tokens_out":23102,"duration_ms":191187,"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":[{"comment":"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.","section":"Theorem 4.2 and Theorem 4.5"},{"comment":"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.","section":"Lemmas 3.10 and 3.11"},{"comment":"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.","section":"Theorem 4.1"}],"minor_comments":[{"comment":"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.","section":"Lemma 3.8"},{"comment":"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.","section":"Proposition 3.6(ii)"},{"comment":"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.","section":"Proposition 3.7(iv)"},{"comment":"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'.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the central idea is promising, and the manuscript does not engage in circular reasoning or parameter fitting. The main obstacle is the missing nonvanishing assertions in Theorems 4.2 and 4.5, together with the incorrect substitution in Lemmas 3.10 and 3.11. If the authors can supply genuine proofs of the needed nonvanishing statements, the paper would be suitable; otherwise the main converses do not follow. I recommend a major revision rather than rejection, because the gaps are localized and may be repairable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper addresses an honest open problem: Serre's 1985 sufficient condition for p26(n)=0, and whether it is necessary. The main results, Theorems 1.4 and 1.5, give conditional 'iff' statements in families where the exponents of primes congruent to 1 mod 12 are constrained mod 5 or 7. This is a genuine partial converse, new relative to Serre and Chan-Cooper-Toh. The strategy—write eta^26 in terms of four CM eigenforms, then control coefficients via congruences mod 5 and 7—is standard but applied with reasonable skill. The paper also gives useful congruence lemmas (Propositions 3.5-3.7) that seem correct.\n\nThe soft spots are real. The most serious is in Theorems 4.2 and 4.5, which are the bridge to the main converse. In Theorem 4.2, after setting t1pm=0, the proof factors t2+(m)+t2-(m) into a parity factor and products over even and odd exponents of primes p=5 mod 12. Even exponents are nonzero by Proposition 3.5, but the odd-exponent product is never shown nonzero. If any t2+(p^odd) vanished, the whole sum would vanish and p26 would be zero, which would break the converse implication. The same gap appears for t1+(p^odd) with p=7 mod 12 in Theorem 4.5. This is not a cosmetic omission; it is the exact step needed to prove p26 != 0.\n\nThere are also fixable errors. Lemma 3.10/3.11 set t2+(p)^2 = 2c (similarly t1+(p)^2 = 2d) but then substitute t2+(p)^(2beta) = 2^(2beta) c^(2beta), which does not follow; the correct relation is (2c)^beta. The conclusion may be recoverable with a 2-adic argument, but the printed algebra is inconsistent. Theorem 4.1 contains a typo: it says alpha is odd for p=5 or 7 mod 12, whereas alpha must be even for 12n+13=1 mod 12; harmless, but it should be corrected.\n\nCitation practice is fine: Serre's decomposition and the Chan-Cooper-Toh formula are used as external input, and the converse direction is not circular. No invented data.\n\nMy overall read: the main idea is sound and the partial converse is likely true, but the paper as printed does not rigorously establish Theorems 1.4/1.5. It deserves a serious referee, not a desk reject—a good referee could tell the authors whether the odd-power nonvanishing can be proven (I suspect it can, using 2-adic valuations, but it is not in the paper). I would not cite it until the gap is closed.","headline":"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.","tokens_in":18706,"tokens_out":7431,"would_cite":false,"duration_ms":64260,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Eta function","vanishing coefficients","Fourier coefficients","Hecke eigenforms","lacunary modular forms","partial converse","Dedekind eta"],"falsifier":"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.","tokens_in":17636,"feed_emoji":"","tokens_out":17998,"duration_ms":159139,"temperature":0.7,"pith_summary":"The eta function raised to the 26th power, $\\eta^{26}$, has coefficients $p_{26}(n)$ that vanish on a set of density zero. Earlier work [7] gave a sufficient condition for a zero and left open whether it was necessary; this paper proves the condition is necessary on two restricted families, arguments $25n+1$ and $49n+3$, under a mild restriction on exponents of primes congruent to $1$ modulo $12$ in $12n+1$. The precise results, Theorem 1.4 and Theorem 1.5, state that in those families $p_{26}(25n+1)=0$ (respectively $p_{26}(49n+3)=0$) exactly when both a prime congruent to $3$ modulo $4$ and a prime congruent to $2$ modulo $3$ divide $12n+1$ to an odd power. A sympathetic reader would care because the paper converts a one-way arithmetic test into an exact description of the zero set for a classical series, giving the first partial converse for the exponent $26$.","feed_headline":"Zeros of eta^26 coefficients get an exact two-prime test","feed_subtitle":"The earlier sufficient condition becomes necessary and sufficient for n=25m+1 and n=49m+3.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the sufficient vanishing criterion and the CM-type Hecke eigenform decomposition, identity (1), on which the whole coefficient analysis rests.","marker":"[7]"},{"why":"Provides the explicit formula for $p_{26}(n)$; the paper's converse is framed as the question left open by that formula and by the earlier theorem.","marker":"[1]"},{"why":"The online calculator used to verify the finite polynomial congruence checks in Propositions 3.5 and 3.7 that support the mod-5 and mod-7 divisibility lemmas.","marker":"[8]"}],"fun_headline_variants":["Eta^26 zero condition is now iff for two families","Converse of Serre's eta^26 criterion proven for 25n+1 and 49n+3","Two-prime test for vanishing eta^26 coefficients: now necessary","Partial converse: eta^26 coefficients vanish exactly when test passes","Eta^26: necessary and sufficient zero test for two progressions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Eta^26 zero condition is now iff for two families","Converse of Serre's eta^26 criterion proven for 25n+1 and 49n+3","Two-prime test for vanishing eta^26 coefficients: now necessary","Partial converse: eta^26 coefficients vanish exactly when test passes","Eta^26: necessary and sufficient zero test for two progressions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000362,"raw_usage":{"total_tokens":1909,"prompt_tokens":857,"completion_tokens":1052,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":952}},"tokens_in":473,"tokens_out":1052,"duration_ms":10947,"temperature":1.0,"reasoning_tokens":952,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:43:20.317549+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Serre, Sur la lacunarit´ e des puissances de η, Glasgow Math","cited_arxiv_id":null,"evidence_quote":"Supplies the sufficient vanishing criterion and the CM-type Hecke eigenform decomposition, identity (1), on which the whole coefficient analysis rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the explicit formula for $p_{26}(n)$; the paper's converse is framed as the question left open by that formula and by the earlier theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The online calculator used to verify the finite polynomial congruence checks in Propositions 3.5 and 3.7 that support the mod-5 and mod-7 divisibility lemmas."}],"review_version":1}