{"id":"41c758ae-ea35-445d-87f3-9578ad32d696","arxiv_id":"2509.07788","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Re(k)>-3, the average of zeta'(1/2+i gamma)^k over zeros is conjectured to be (1/Gamma(k+2)) (log(T/2pi))^k.","lead":"The paper conjectures a simple Gamma-function formula for the complex moments of the derivative of the Riemann zeta function at its zeros, for all powers with real part above -3. It derives this prediction from two independent random matrix computations and proves the first-moment case.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Splitting Conjecture (pseudo-independence of P_X and Z_X) is assumed for all Re(k)>-3 but verified only at k=1; if it fails at any other k, Conjecture 1's constant is wrong.","rationale":"The paper is a conjecture-generation paper: the random-matrix results are rigorous, but the leap to zeta depends crucially on the Splitting Conjecture. The reader's CONDITIONAL verdict is appropriate. My stress-test finds no internal inconsistency in the derivations beyond the small repairable issue in equation (6.2) already noted; the central soft spot is the unproved splitting for general k, which the paper itself acknowledges. A numerical k=2 check is feasible and would materially inform confidence, but it does not change the logical status: the claim remains a well-motivated conjecture conditional on the splitting hypothesis. Hence no verdict adjustment is needed.","tokens_in":13336,"tokens_out":12538,"duration_ms":124187,"concrete_test":"Evaluate the k=2 discrete derivative moment numerically: for T=10^6 (or 10^7), compute S_2(T) = (1/N(T)) sum_{0<γ≤T} ζ'(1/2+iγ)^2 and compare with (log(T/2π))^2/12. Since k=2 is an integer, no branch definition is required, isolating the product step from the branch issue. If S_2(T) differs from the predicted value by more than the expected error (or appears to have a different log-power), the Splitting Conjecture fails for k=2 and Conjecture 1 is refuted. Agreement would provide the first evidence beyond k=1 that the factorization survives for unproved k. To also probe non-integer k, the same check with k=1/2 using the hybrid-product branch defined in Section 3 would test the branch choice.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 states: \"It is believed that P_X(s) and Z_X(s) operate pseudo-independently ... so the moments of zeta are products of moments of P_X and Z_X\" and \"We justify ... in the case k=1 in Theorem 12.\" This is the load-bearing step: Theorems 4 and 6 compute the Z'_X factor (and the characteristic-polynomial analogue), Theorem 7 computes the P_X factor, but the conjecture requires multiplying these averages for every k with Re(k)>-3. The paper proves this factorization only for k=1 (Theorem 12); k=-1 is cited from the literature but no internal verification is given, and no other k is checked. If the pseudo-independence fails at, say, k=2 — meaning the averages of P_X(ρ)^2 and Z'_X(ρ)^2 do not multiply to the average of their product — then the leading constant in Conjecture 1 would acquire an arithmetic correction, precisely the failure mode the hybrid model was introduced to rule out. This is not a defect internal to the RMT computations; it is an unproved heuristic. The paper's own language ('It is believed') flags it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper formulates Conjecture 1, asserting that for Re(k)>-3 the discrete complex moments of zeta'(rho) at zeros satisfy (1/N(T)) sum_{0<gamma<=T} zeta'(1/2+i gamma)^k ~ (1/Gamma(k+2)) (log(T/2pi))^k as T->infty, with a branch convention based on the hybrid Euler-Hadamard product. The support consists of two independent random matrix calculations: Theorem 4 uses Selberg's integral for the derivative of unitary characteristic polynomials, and Theorem 6 uses a Toeplitz-determinant/Fisher-Hartwig computation for the hybrid model Z'_X. Theorem 7 shows that the average of P_X(rho)^k is 1 for complex k under RH, and Theorem 12 uses a twisted first moment to verify the resulting factorization for k=1 only.","tokens_in":13654,"tokens_out":6035,"duration_ms":63270,"significance":"If Conjecture 1 is correct, it provides the first conjectured leading-order asymptotics for complex moments of zeta' at its zeros, covering all Re(k)>-3 and unifying the known k=1 and k=-1 cases. The two random matrix computations are rigorous and mutually consistent, and they give the same constant and the same analytic region Re(k)>-3. The proof that the P_X factor contributes a mean of 1 for every complex k is a valuable independent result. The paper is also transparent about the role of the unproved Splitting Conjecture, although the main conjecture depends on it essentially.","major_comments":[{"comment":"The passage 'It is believed that P_X(s) and Z_X(s) operate pseudo-independently ... so the moments of zeta are products of moments of P_X and Z_X' is the load-bearing step. To obtain Conjecture 1 for every k with Re(k)>-3, one must multiply the averages from Theorem 6 and Theorem 7, i.e. assume E[P_X(rho)^k Z'_X(rho)^k] = E[P_X(rho)^k] E[Z'_X(rho)^k] for all such k. The only internal verification is k=1 (Theorem 12); the k=-1 case is cited from the literature but is not shown to follow from the same splitting, and no other k is tested. If the factorization fails for some k, say k=2, the leading constant in Conjecture 1 would acquire an arithmetic correction. This is not a flaw in the RMT derivations themselves, but it means Conjecture 1 as stated is stronger than the paper's evidence. The conjecture should either be stated as conditional on the Splitting Conjecture, or additional checks","section":"Section 3, after Theorem 7"},{"comment":"The complex power zeta'(rho)^k is not well-defined without a branch choice, and the paper explicitly notes that the appropriate branch is 'not the same as that which comes from continuously varying log zeta'(s)' and is instead tied to the hybrid product. However, Conjecture 1 itself never states this branch convention. For non-integer k, a different branch changes the left side by a phase e^{2 pi i n k}, while the right side has a fixed phase from (log(T/2pi))^k. Since the conjecture is central, the branch definition (e.g. via the partial Hadamard product Z_X and the limiting procedure described in Section 3) must be part of the conjecture statement, not only an informal remark preceding it.","section":"Section 1 and Conjecture 1"}],"minor_comments":[{"comment":"The proof begins 'In the case when Re(s)=1/2 and X=(log T)^{2-epsilon}', which is broader than the theorem's assumption X=O(log T). This is not a substantive problem, but the relation between the two choices of X should be clarified so the reader sees why the O(log T) condition is sufficient.","section":"Proof of Theorem 7, Section 6"},{"comment":"Typo: 'eignenangles' should be 'eigenangles'.","section":"Theorem 6 statement"},{"comment":"The remark after the proof states the result holds for k not in {-3,-4,-5,...}. This set appears to omit k=-2, where Gamma(k+2) has a pole but the leading asymptotic is interpreted as zero. A sentence explaining the limiting interpretation at k=-2 would remove ambiguity.","section":"Section 5, proof of Theorem 6"},{"comment":"The displayed formula for N(T) is easy to misread. Consider adding parentheses: N(T) = (T/(2pi)) log(T/(2pi e)) + O(log T).","section":"Conjecture 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid random-matrix-theory contribution and the two independent computations are a genuine strength. My main reservation is that the headline Conjecture 1 depends on the Splitting Conjecture for a continuum of complex k, with only k=1 verified internally. That is a standard type of heuristics in this area, but the conjecture should be framed accordingly. The branch issue is also worth tightening. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does something genuinely new: it predicts the leading-order asymptotics for all complex moments of zeta'(rho) for Re(k)>-3, with no arithmetic factor at that order. The random matrix side is solid: the Selberg integral evaluation (Theorem 4) and the Toeplitz computation (Theorem 6) are independent and agree, and the k=1 case is anchored by a twisted first moment (Theorem 12). If Conjecture 1 holds, it stitches together the proven k=1 and conditional k=-1 cases and gives a clean prediction for all other k.\n\nThe two random matrix calculations are the real content. Theorem 4 is an exact formula for complex moments of Z'(theta) from Selberg's integral; Theorem 6 handles the hybrid Z_X via Toeplitz determinants. They cross-check each other, which is strong evidence for the RMT prediction. The paper is also honest about the arithmetic factor: the hybrid model naturally predicts none at leading order for complex moments, a nice contrast to the |zeta'| case.\n\nThe soft spot is exactly where you would expect: the Splitting Conjecture. The paper assumes P_X and Z'_X are pseudo-independent for every k with Re(k)>-3, but verifies the factorization only for k=1 (Theorem 12). If that independence fails at some other k, the constant in Conjecture 1 gets an arithmetic correction. The paper flags this openly ('It is believed...'), so it is not hiding the heuristic. It is the difference between a conjecture and a theorem, and the title says conjecture. The branch definition for zeta'(rho)^k is subtle and handled via the hybrid product; it is discussed, though a referee will want it made fully explicit. One genuine but repairable slip: in the proof of Theorem 7, inequality (6.2) writes the sum of d_{|k|}(m) over all m as equal to P_X(0)^{|k|}; that sum diverges. The intended bound is sum of |a_k(m)| = P_X(0)^{|k|}, which is finite, so the error estimate survives. Minor.\n\nThis is a paper for analytic number theorists and random matrix people. It deserves a serious referee even though the main zeta statement is conjectural; the RMT results are new and the conjecture is likely to shape the next few papers in this area. I would send it to peer review.","headline":"New general conjecture for complex moments of zeta'(rho) with two clean random matrix derivations; the load-bearing splitting heuristic is the only real caveat.","tokens_in":14146,"tokens_out":4628,"would_cite":true,"duration_ms":44216,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","11M50"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper conjectures that complex moments of the derivative of the Riemann zeta function at its non-trivial zeros follow a single asymptotic formula for every exponent with real part greater than −3.","keywords":["Riemann zeta function","derivative at zeros","complex moments","discrete moments","random matrix theory","hybrid product of prime and zero factors","Toeplitz determinants","gamma function"],"falsifier":"Numerically test the normalised sum for a non-integer k with Re(k)>−3, say k=1/2, using a large set of zeros up to height T and the branch defined by the hybrid product; if the ratio to (log(T/2π))^{1/2} / Γ(5/2) does not approach 1 as T grows, the conjecture or the splitting heuristic fails. A cheaper check is to examine k=−1/2 or k=3/2, since the k=1 case is already proven.","tokens_in":13256,"feed_emoji":"🧮","tokens_out":10337,"duration_ms":112102,"temperature":0.7,"pith_summary":"This paper proposes a leading-order formula for the complex moments of the derivative of the Riemann zeta function evaluated at its zeros. For every complex exponent k with real part greater than −3, the normalised sum over zeros is conjectured to approach (log(T/2π))^k / Γ(k+2), using the branch of the complex power fixed by the hybrid product of prime and zero factors. This is the first general conjecture for these complex moments, and it reproduces the two known cases k=1 and k=−1. Its notable prediction is that no arithmetic factor survives at leading order, in contrast to the corresponding absolute-value moments. The evidence comes from two independent random-matrix computations and a proof that the underlying splitting assumption holds for k=1.","feed_headline":"Paper predicts one formula for complex moments of ζ′ at zeros","feed_subtitle":"If right, every complex k with Re(k)>−3 averages to (log T/2π)^k/Γ(k+2), with no arithmetic factor.","key_machinery":"The central mechanism is the hybrid factorisation ζ(s) ≈ P_X(s) Z_X(s), where P_X is a truncated product over primes and Z_X is a truncated product over zeros. Differentiating at a zero gives ζ′(ρ) ≈ P_X(ρ) Z′_X(ρ), and the pseudo-independence of P_X and Z_X, the Splitting Conjecture, turns the moment of ζ′(ρ)^k into the product of the moment of P_X(ρ)^k and the moment of Z′_X(ρ)^k. The paper evaluates those two moments by exact methods: a multi-dimensional beta integral gives the characteristic-polynomial analogue, a Toeplitz determinant with one singularity at the origin gives the zero-factor analogue, and a zero-sum formula gives the prime-factor mean. Both factor analogues produce N^k /","core_discovery":"On the paper's own terms, the central claim is Conjecture 1: assuming the Riemann Hypothesis, for Re(k)>−3, the average of ζ′(1/2+iγ)^k over zeros γ with 0<γ≤T is asymptotically (log(T/2π))^k / Γ(k+2). The derivation shows that a Haar-averaged unitary characteristic polynomial has exact complex derivative moments e^{iπk/2} Γ(N+k+1) / (N! Γ(k+2)), asymptotic to e^{iπk/2} N^k / Γ(k+2); the same leading constant, without the factor i^k once θ-differentiation is translated to t-differentiation, is obtained for the zero factor in the hybrid model. The prime factor separately has average 1 over the zeros. The branch is not obtained by continuous variation of log ζ′(s), but by the product represent","pith_inferences":["The proof of the zero-factor theorem depends on the Fourier coefficients of the smoothing factor vanishing for all non-positive modes, so the leading constant should be independent of the particular smoothing function in the hybrid product; testing two different smoothings numerically would probe universality.","If the Splitting Conjecture fails, the first symptom would be a missing lower-order constant in the averaged ratio, not a visible failure at k=1; computing the normalised sum for k=1/2 at increasing heights is a sharper test than the k=1 verification.","The same gamma-only leading constant would plausibly carry over to other L-functions with unitary symmetry, where the discrete zeros play the role of eigenangles; the paper does not state this, but the mechanism is generic."],"forward_implications":["Every exponent Re(k)>−3 is covered by one closed formula; no new arithmetic constant is introduced.","The k=−2 case is predicted to have a vanishing leading term, with a heuristic bound O(T^{−1/2+ε}); this zero is a direct consequence of the pole of the gamma factor at k=−2.","The known k=1, proven unconditionally, and k=−1, known conditionally on simple zeros, arise as special cases, so the conjecture interpolates the existing evidence.","Because ζ′(ρ) is complex, numerical or analytic tests must adopt the hybrid-product branch rather than continuous variation of log ζ′(s); the paper identifies this as the correct convention.","The same hybrid-plus-random-matrix route is announced as extending to mixed products of higher derivatives of ζ at zeros."],"supporting_citations":[{"why":"Supplies the random-matrix model identifying the characteristic polynomial of a unitary matrix with the value distribution of zeta near height T.","marker":"[11]"},{"why":"Establishes the discrete-moment analogue for absolute values of the derivative of the characteristic polynomial, which this paper adapts to complex powers.","marker":"[9]"},{"why":"Provides the hybrid product formula for zeta and formulates the Splitting Conjecture that lets the moments factor.","marker":"[7]"},{"why":"Extends the hybrid method to discrete moments of the derivative at zeros and supplies the differentiated hybrid formula and the prime-factor bound used here.","marker":"[2]"},{"why":"Is the source of the exact multi-dimensional beta integral used to evaluate the characteristic-polynomial moment in Theorem 4.","marker":"[12]"},{"why":"Gives the Toeplitz-determinant asymptotic with one singularity at the origin that yields the leading constant in Theorem 6.","marker":"[4]"},{"why":"Supplies the twisted first moment of the derivative at zeros used to prove the k=1 splitting case in Theorem 12.","marker":"[1]"},{"why":"Is the known unconditional k=1 result that Conjecture 1 must reproduce.","marker":"[3]"},{"why":"Supplies the known k=−1 case, conditional on simple zeros, that Conjecture 1 must reproduce.","marker":"[5]"},{"why":"Provides the uniform zero-sum formula used to show that the prime-factor moment averages to 1 in Theorem 7.","marker":"[6]"}],"fun_headline_variants":["New conjecture for ζ′ moments at zeros","ζ′ moments at zeros: one clean formula","Formula predicts ζ′ moments at all zeros","Conjecture: simple asymptotic for ζ′ at zeros","Zero moments of ζ′: a single universal formula"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The conjecture stands on the pseudo-independence of the truncated prime product P_X and the truncated zero product Z_X at the zeros of zeta; this splitting has been verified only for k=1, and if it fails for another k with Re(k)>−3 the formula would be wrong even though the random-matrix computations are individually correct.","fun_headline_variants_meta":{"raw":{"variants":["New conjecture for ζ′ moments at zeros","ζ′ moments at zeros: one clean formula","Formula predicts ζ′ moments at all zeros","Conjecture: simple asymptotic for ζ′ at zeros","Zero moments of ζ′: a single universal formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000191,"raw_usage":{"total_tokens":1136,"prompt_tokens":657,"completion_tokens":479,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":401,"completion_tokens_details":{"reasoning_tokens":407}},"tokens_in":401,"tokens_out":479,"duration_ms":5459,"temperature":1.0,"reasoning_tokens":407,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T21:43:54.770235+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically test the normalised sum for a non-integer k with Re(k)>−3, say k=1/2, using a large set of zeros up to height T and the branch defined by the hybrid product; if the ratio to (log(T/2π))^{1/2} / Γ(5/2) does not approach 1 as T grows, the conjecture or the splitting heuristic fails. A cheaper check is to examine k=−1/2 or k=3/2, since the k=1 case is already proven.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the random-matrix model identifying the characteristic polynomial of a unitary matrix with the value distribution of zeta near height T."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the discrete-moment analogue for absolute values of the derivative of the characteristic polynomial, which this paper adapts to complex powers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the hybrid product formula for zeta and formulates the Splitting Conjecture that lets the moments factor."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the hybrid method to discrete moments of the derivative at zeros and supplies the differentiated hybrid formula and the prime-factor bound used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the source of the exact multi-dimensional beta integral used to evaluate the characteristic-polynomial moment in Theorem 4."},{"cited_title":"Ehrhardt and B","cited_arxiv_id":null,"evidence_quote":"Gives the Toeplitz-determinant asymptotic with one singularity at the origin that yields the leading constant in Theorem 6."},{"cited_title":"Benli, E","cited_arxiv_id":null,"evidence_quote":"Supplies the twisted first moment of the derivative at zeros used to prove the k=1 splitting case in Theorem 12."},{"cited_title":"Jean Coquet","cited_arxiv_id":null,"evidence_quote":"Is the known unconditional k=1 result that Conjecture 1 must reproduce."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the known k=−1 case, conditional on simple zeros, that Conjecture 1 must reproduce."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the uniform zero-sum formula used to show that the prime-factor moment averages to 1 in Theorem 7."}],"review_version":1}