{"id":"cb3af657-6095-4f62-9de4-5c5095ae2925","arxiv_id":"2509.06390","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Higher logarithmic-derivative moments at s=1 over Dirichlet characters of prime modulus converge to an explicit series, but the advertised exceptional-zero application is missing from the body.","lead":"This paper studies averages of higher derivatives of the logarithmic derivative of Dirichlet L-functions at s=1, as the character varies over large prime conductors. It derives explicit moment asymptotics, but the abstract's promised application to exceptional zeros in cyclotomic fields is not present in the manuscript.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1's exact orthogonality identity (25) is false for the non-principal character average; the missing principal-character bias term invalidates the proof of Proposition 3.2 and hence Theorems 1.2–1.3 as written.","rationale":"The reader's weakest_assumption correctly identifies the false orthogonality identity (25) in Lemma 3.1. This is load-bearing because Proposition 3.2 and the subsequent moment theorems rely on it. My independent check of the character sum shows the missing bias term is -S_g^{a+b}/(φ(m)-1); for the specific g this is O((log x)^{(r+1)(a+b)}/m), which lies within the claimed error, so the theorem may be repairable. Nevertheless, the paper as submitted contains a false lemma in a key step, and the abstract's exceptional-zero application is nowhere proved in the visible text. Both issues support the REJECT verdict. I do not see additional independent support that would overturn this: Theorem 1.1 is a standard explicit formula, and the moments may hold after correction, but a false identity cannot be accepted without revision.","tokens_in":13494,"tokens_out":9325,"duration_ms":96806,"concrete_test":"Re-derive Proposition 3.2 using the exact non-principal orthogonality relation to compute the bias term explicitly, and verify whether the bias satisfies the claimed O((log x)^{(r+1)(a+b)+2}/m) error. As a numerical spot-check, compute both sides of (25) for m=5, g(2)=g(3)=1, all other g=0, a=b=1; the left side over non-principal characters is 4/3 while the right side is 2, confirming the discrepancy.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.1's identity (25) is false for the averaging set used. For prime m, the sum over all φ(m) characters satisfies Σ_χ χ(n)\\bar{χ}(n') = φ(m) 1_{n≡n'}, but restricting to non-principal characters changes this to φ(m)1_{n≡n'} − 1. Hence the correct form of (25) is (φ(m)/(φ(m)−1)) Σ_{j=1}^{m−1} λ^{(a)}λ^{(b)} − (1/(φ(m)−1)) S_g^{a+b}, where S_g = Σ_{n<x} g(x,n). For the g chosen in Proposition 3.2, the missing bias term is O((log x)^{(r+1)(a+b)}/m), which is smaller than the claimed error O((log x)^{(r+1)(a+b)+2}/m), so the main term may survive a corrected proof. However, the proof as written uses the false exact identity, so Theorem 1.2 and 1.3 are not rigorously established. The abstract's advertised application (Li-coefficient positivity and exceptional-zero exclusion) is also completely absent from the manuscript text, so the headline claim is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the averages of monomials P^{(a,b)}(z)=z^a\\bar z^b of the higher logarithmic derivatives \\mathcal L^{(r)}(1,\\chi)= (L'/L)^{(r)}(1,\\chi) of Dirichlet L-functions attached to non-principal characters modulo a large prime m. Theorem 1.1 gives an arithmetic limit formula for \\mathcal L^{(r)}(1,\\chi) in terms of a truncated prime-power sum. Theorems 1.2 and 1.3 assert, under GRH and unconditionally respectively, that the (a,b)-moment equals (-1)^{(r+1)(a+b)} \\mu^{(a,b)}(r)+O(m^{\\varepsilon-1}), where \\mu^{(a,b)}(r) is an explicit convergent series. The abstract additionally claims an application to positivity of the second Li coefficient and exclusion of an exceptional zero in a Stark-type region, but this application does not appear in the text.","tokens_in":13888,"tokens_out":10574,"duration_ms":112574,"significance":"If the moment theorems are established, they would naturally extend the Ihara--Murty--Shimura first-logarithmic-derivative moment result to all derivatives and provide an unconditional higher-derivative moment asymptotic at the edge of the critical strip. The constant \\mu^{(a,b)}(r) is an intrinsic convergent Euler-type sum with no fitted parameters, and the strategy via an explicit arithmetic formula plus standard zero-density estimates is appropriate. The abstract's advertised application to Li coefficients would add arithmetic significance, but it is not present in the manuscript. However, the central computational lemma contains a correctness gap that currently invalidates the proofs of the main moment results as written.","major_comments":[{"comment":"The claimed exact identity is false for the average over non-principal characters. Orthogonality over the full group gives (1/\\varphi(m))\\sum_\\chi \\chi(n)\\bar\\chi(n')=1_{n\\equiv n'}; after excluding \\chi_0 the right-hand side becomes (\\varphi(m)/(\\varphi(m)-1))\\sum_{j=1}^{m-1}\\lambda^{(a)}\\lambda^{(b)} - (1/(\\varphi(m)-1))S_g^{a+b}, where S_g=\\sum_{n<x}g(x,n). The proof's assertion that 'when summed over all \\chi' only congruent products contribute omits the principal-character subtraction. Since Eq (27) and hence Proposition 3.2 use (25), the proofs of Theorems 1.2 and 1.3 are not rigorous as written. The missing bias is O((\\log x)^{(r+1)(a+b)}/m) for the g chosen in Prop 3.2, which is smaller than the claimed error, so a corrected proof is plausible; but the current statement is false.","section":"Lemma 3.1, Eq (25)"},{"comment":"The abstract promises that the moment asymptotics imply positivity of the second Li coefficient of prime cyclotomic fields and rule out an exceptional zero in a Stark-type zero-free region. The visible manuscript ends with the proof of Theorem 1.3; there is no definition of Li coefficients, no Stark-type region, and no application. This unsupported claim must either be supplied as a genuine section or removed from the abstract.","section":"Abstract and text"},{"comment":"After writing \\lambda^{(k)}=\\Lambda_{r,k}/j+E_k with E_k=O((\\log x)^{(r+1)k+1}/m), the displayed estimate\n1/|X_m|\\sum P = \\sum_j \\Lambda_{r,a}\\Lambda_{r,b}/j^2 + O((\\log x)^{(r+1)(a+b)+2}/m^2)\nis not justified: the linear cross terms \\sum_j (\\Lambda_{r,a}/j)E_b + E_a(\\Lambda_{r,b}/j) are of size O((\\log x)^{(r+1)(a+b)+1}/m), not /m^2. This is still smaller than the proposition's final error, so the claim may survive, but the intermediate estimate is incorrect as written. Similarly, Proposition 3.8(2) states O((\\log x)^{16}) while its proof concludes O((r+1)!2^r(\\log x)^{2r+15}); the statement and proof need to be reconciled.","section":"Proposition 3.2, error analysis"}],"minor_comments":[{"comment":"The second factor in the product g_\\chi(x)^a g_\\chi(x)^b should be written with a conjugate bar, \\overline{g_\\chi(x)}^b, to match the definition of P^{(a,b)}(z)=z^a\\bar z^b.","section":"Lemma 3.1"},{"comment":"The discussion about including the principal character is confusing, since X_m is already defined as the set of non-principal characters; the paragraph should be removed or clarified.","section":"Proposition 3.2, final paragraph"},{"comment":"The attribution 'Paley and Selberg' is followed by a reference only to Paley; a Selberg reference should be added or the name adjusted.","section":"Introduction"},{"comment":"The error term O((\\log x)^{r+2}/x) in (34) is dominated by the first error term for x=m^2; the display could be simplified to avoid the impression of a missing factor.","section":"Section 3.1, Eq (34)"},{"comment":"There are occasional grammatical slips (e.g., 'will effect' should be 'will affect'), and some displayed expressions have missing braces (e.g., around \\Lambda_{r,k} in the introduction).","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main moment result is likely salvageable: the orthogonality identity can be corrected, and the extra principal-character bias term is smaller than the target error. But the current version's central lemma is false and the advertised application is missing, so the paper is not acceptable in its present form. I recommend asking the author to fix Lemma 3.1 and the error analysis, and to either add the Li-coefficient/exceptional-zero application or withdraw it from the abstract. The paper fits the scope of a number theory journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good seeing you at the workshop. Quick take on arXiv:2509.06390.\n\nThe genuine content here is Theorems 1.2 and 1.3: for r≥1, moment asymptotics for L^(r)(1,χ) over non-principal characters of prime conductor, with an unconditional error m^{ε−1}. That is a real extension of Ihara–Murty–Shimura's r=0 result, and the author works through the zero-density machinery cleanly. The constant μ^{(a,b)}(r) is well-defined and no parameters are fitted. I believe the moment formulas are probably true.\n\nBut the proof as written has a hole in the load-bearing wall. Lemma 3.1's identity (25) asserts that averaging g_χ^a ḡ_χ^b over non-principal characters is exactly the sum over j of λ^{(a)}(j)λ^{(b)}(j). That is only true with full orthogonality including the principal character. Restricting to X_m^* gives an extra −(φ(m)−1)^{-1} S_g^{a+b} bias term (with S_g = Σ_{n<x} g(x,n)). The stress-test note says that for the specific g used in Proposition 3.2 the extra term is O((log x)^{(r+1)(a+b)}/m), which is smaller than the displayed error O((log x)^{(r+1)(a+b)+2}/m), so the main term might survive a corrected proof. But as printed, the identity is false and Proposition 3.2 is not established. That's not a nitpick; the moment theorems 1.2–1.3 rest on it.\n\nSecond issue: the abstract promises a Li-coefficient positivity result and an exceptional-zero exclusion in a Stark-type zero-free region. The visible text contains no such application. The paper ends with the proof of Theorem 1.3. So the advertised headline is unsupported. Either the application is in a companion part of the thesis or it was cut accidentally. The version I see does not contain it.\n\nOn the positive side, the paper credits prior work (Ihara–Murty–Shimura, the author's own [4]) and does not overreach in the technical parts. The zero-density estimates and the treatment of the unique exceptional character look standard. I would not call the work circular; [4] is a preprint, but the dependence is explicit.\n\nWho is this for? Analytic number theorists working on distribution of L'/L values and the Ihara–Murty–Shimura line. It is a meaningful incremental result if the identity is fixed and the application is either supplied or removed from the abstract.\n\nMy advice: send it to a careful referee, not desk reject. The moment theorems are likely correct and worth referee time. But tell the referee to check Lemma 3.1 carefully, and ask the author to provide the missing application or revise the claim. If the identity cannot be fixed, then reject; but I suspect it can.","headline":"A real extension of IMS moments that is currently undermined by a false orthogonality identity and an absent advertised application.","tokens_in":14294,"tokens_out":2587,"would_cite":false,"duration_ms":27623,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M20","11R18","11R42"],"pacs":[],"model":"deepseek-v4-flash","headline":"Averages of higher logarithmic-derivative moments of Dirichlet L-functions at s=1 converge unconditionally to explicit constants, and this rules out an exceptional zero for large prime cyclotomic fields.","keywords":["Dirichlet L-functions","logarithmic derivative","moments","higher derivatives","cyclotomic fields","Li coefficients","exceptional zeros","character orthogonality"],"falsifier":"Evaluate the two sides of identity (25) for a small prime conductor, say m=7, with a=b=1 and a simple test function such as g(x,n)=1 for x=10; any non-zero difference between the average over the six non-principal characters and Σ_{j=1}^6 λ^(1)(j)λ^(1)(j) shows the lemma is false as stated. For the final theorem, numerically compute (1/|X_m|)Σ P^(1,1)(L^(1)(1,χ)) for primes up to a few hundred and check whether the deviation from μ^(1,1)(1) decays like O(m^{−1+ε}); a slower decay would contradict Theorem 1.3.","tokens_in":13455,"feed_emoji":"🔢","tokens_out":10230,"duration_ms":107675,"temperature":0.7,"pith_summary":"This paper studies the moments of higher derivatives of the logarithmic derivative of Dirichlet L-functions at s=1. For each order r and each pair of exponents (a,b), it proves that the average of P^(a,b)(L^(r)(1,χ)) over all non-principal characters modulo a large prime conductor m converges, unconditionally, to an explicitly computable constant μ^(a,b)(r), with error O(m^{ε−1}) for any ε>0. This extends the r=0 result of Ihara, Murty, and Shimura to all derivative orders. The author uses this moment control to argue that the second Li coefficient of a prime cyclotomic field of large conductor is positive and, combined with a zero-free criterion, to exclude a possible exceptional zero in a Stark-type zero-free region for these fields. A reader should care because these moment asymptotics describe how the values of L'/L near the central point distribute over a family of characters, and the zero-free consequence targets a classical exceptional-zero question in cyclotomic fields.","feed_headline":"Higher log-derivative moments converge unconditionally","feed_subtitle":"Averages over prime-conductor characters hit an explicit constant with error m^{ε−1}, shrinking faster than any power.","key_machinery":"The engine is the arithmetic function Φ(χ,r,x) = (x−1)^{-1} Σ_{n<x} (x/n−1)χ(n)Λ(n)(log n)^r, which Theorem 1.1 identifies with L^(r)(1,χ) up to sign as x→∞. To compute moments, the paper averages Φ(χ,r,x) first; a character-orthogonality lemma (Lemma 3.1) turns the product of two such character sums into a sum over residue classes j modulo m of convolutions λ^(a)(j,x)λ^(b)(j,x). The main term is the tail of Σ_j Λ_{r,a}(j)Λ_{r,b}(j)/j², while the error comes from bounding the long-residue-class shifts and, for the unconditional passage, from estimating sums of x^β over zeros of L(s,χ).","core_discovery":"The central claim is Theorem 1.3: for fixed nonnegative integers a,b and r≥0, the average (1/|X_m|) Σ_{χ∈X_m} P^(a,b)(L^(r)(1,χ)) equals (−1)^{(r+1)(a+b)} μ^(a,b)(r) + O_{r,a,b}(m^{ε−1}) for every ε>0, so the limit holds unconditionally as m→∞. The constant is μ^(a,b)(r)=Σ_{j≥1} Λ_{r,a}(j)Λ_{r,b}(j)/j², built from k-fold Dirichlet convolutions of Λ(n)(log n)^r. The paper first derives an arithmetic formula (Theorem 1.1) expressing L^(r)(1,χ) as the limit of a weighted sum over prime powers, then computes the moments of that arithmetic sum under GRH via character orthogonality, and finally removes the GRH assumption using zero-density and zero-free-region estimates, treating the possible exce","pith_inferences":["A corrected orthogonality identity—one that accounts for the bias term when products are divisible by the conductor—would likely preserve the main theorem, since the missing contribution is of order 1/(φ(m)−1) and should fall inside the O(m^{ε−1}) error.","The same machinery should extend to mixed moments such as P^(a,b)(L^(r)(1,χ)L^(s)(1,χ)), giving the joint distribution of the derivative vector (L^(0)(1,χ), L^(1)(1,χ), …).","For composite conductors the character orthogonality is messier, but an explicit bias term can be written down, suggesting the moment limit should persist for conductors tending to infinity without the primality restriction.","If the Li-coefficient criterion can be made effective, the method could quantify how large the conductor must be to exclude the exceptional zero, turning a qualitative statement into an explicit one."],"forward_implications":["The unconditional moment limit holds for every fixed r,a,b: the averaged P^(a,b)(L^(r)(1,χ)) converges to (−1)^{(r+1)(a+b)} μ^(a,b)(r) as the prime conductor goes to infinity.","The error O(m^{ε−1}) has the same quality as the known r=0 case, so higher-derivative moments are controlled at essentially the same strength as the base case.","For r=0 the theorem recovers the Ihara–Murty–Shimura moment result with a uniform unconditional error term of the form O(m^{ε−1}).","According to the paper's stated application, positivity of the second Li coefficient for prime cyclotomic fields of large conductor follows, and together with a zero-free criterion this rules out an exceptional zero in a Stark-type zero-free region.","The arithmetic formula for L^(r)(1,χ) itself gives explicit finite approximations at scale x that are usable in numerical evaluations of these higher derivatives."],"supporting_citations":[{"why":"Provides the r=0 moment theorem, the arithmetic formula for L'(1,χ)/L(1,χ), and the orthogonality lemmas this paper generalizes.","marker":"[18]"},{"why":"Supplies the contour-integration template for converting the logarithmic-derivative expansion into an arithmetic limit formula.","marker":"[4]"},{"why":"Gives the functional equation, Theorems (A) and (B) on zeros, and the zero-counting bound used in the unconditional estimates.","marker":"[3]"},{"why":"Stark's lemma yields the Hadamard-product form of L'/L that the derivative expansions start from.","marker":"[17]"},{"why":"Montgomery's zero-density theorem is the input for Lemma 3.5 bounding sums of x^β over zeros.","marker":"[13]"},{"why":"Supplies the definitions and bounds for the arithmetic functions Λ_k and Λ_{r,k}.","marker":"[8]"}],"fun_headline_variants":["Unconditional moment asymptotics for higher log derivatives","L-function log derivatives: moments converge unconditionally","Exceptional zero excluded via Li coefficient positivity","Second Li coefficient positive for large prime cyclotomics"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The moment proof depends on an orthogonality identity for non-principal characters; as written it omits a correction term that standard orthogonality predicts, and if that omission is real the proofs of the moment theorems need adjustment, even though the stated limits may still be true.","fun_headline_variants_meta":{"raw":{"variants":["Unconditional moment asymptotics for higher log derivatives","L-function log derivatives: moments converge unconditionally","Exceptional zero excluded via Li coefficient positivity","Second Li coefficient positive for large prime cyclotomics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1366,"prompt_tokens":766,"completion_tokens":600,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":541}},"tokens_in":510,"tokens_out":600,"duration_ms":7921,"temperature":1.0,"reasoning_tokens":541,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T23:42:15.341845+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the two sides of identity (25) for a small prime conductor, say m=7, with a=b=1 and a simple test function such as g(x,n)=1 for x=10; any non-zero difference between the average over the six non-principal characters and Σ_{j=1}^6 λ^(1)(j)λ^(1)(j) shows the lemma is false as stated. For the final theorem, numerically compute (1/|X_m|)Σ P^(1,1)(L^(1)(1,χ)) for primes up to a few hundred and check whether the deviation from μ^(1,1)(1) decays like O(m^{−1+ε}); a slower decay would contradict Theorem 1.3.","supporting_citations":[{"cited_title":"Kumar Murty,On the logarithmic derivatives of Dirichlet L-functions at s=1, Acta Arithmetica137(2009), no","cited_arxiv_id":null,"evidence_quote":"Provides the r=0 moment theorem, the arithmetic formula for L'(1,χ)/L(1,χ), and the orthogonality lemmas this paper generalizes."},{"cited_title":"Higher Euler-Kronecker Constants of Number fields","cited_arxiv_id":"2411.17946","evidence_quote":"Supplies the contour-integration template for converting the logarithmic-derivative expansion into an arithmetic limit formula."},{"cited_title":"Davenport,Multiplicative Number Theory, Third edition, Springer Verlag, New York, 2000","cited_arxiv_id":null,"evidence_quote":"Gives the functional equation, Theorems (A) and (B) on zeros, and the zero-counting bound used in the unconditional estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Stark's lemma yields the Hadamard-product form of L'/L that the derivative expansions start from."},{"cited_title":"227, Springer Verlag, 1971","cited_arxiv_id":null,"evidence_quote":"Montgomery's zero-density theorem is the input for Lemma 3.5 bounding sums of x^β over zeros."},{"cited_title":"M-functions","cited_arxiv_id":null,"evidence_quote":"Supplies the definitions and bounds for the arithmetic functions Λ_k and Λ_{r,k}."}],"review_version":1}