{"id":"35844102-3d2c-42bc-a9fa-1c9168671083","arxiv_id":"2509.07792","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New conjectures give the full asymptotic expansion of discrete moments of mixed derivatives of the Riemann zeta function at its zeros, including a previously unknown formula for the second moment of ζ'(ρ).","lead":"This paper conjectures complete asymptotic formulas for sums of products of derivatives of the Riemann zeta function evaluated at its zeros, derived from the Ratios Conjecture and random matrix theory. If correct, it settles the next open step in the Shanks conjecture program and yields a new, numerically tested formula for the second moment of the first derivative of zeta.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Holomorphy of the main term in the shift parameters is cited but not demonstrated; a surviving pole at α_j=0 or α_j=α_ℓ would invalidate the derivative-moment conjectures and the evaluation in Conjecture 1.","rationale":"The paper is explicitly conjectural, and the diagonal-dominance step of the Ratios recipe is the genre's accepted heuristic; the authors are transparent about it. The more pointed and checkable gap is the holomorphy of the main term in the shift parameters, which Remark 5 flags as non-obvious and then dismisses with a citation. That property is load-bearing because every derivative moment in Conjectures 3–5 is obtained by Taylor expansion and term-by-term differentiation in α_j; if the main term has a pole at α_j=0 or α_j=α_ℓ, those operations are invalid even if the recipe's main term is correct. The reader identified this as a secondary premise; I elevate it to the primary concern because it is a concrete mathematical condition that can be verified from (8.1) and (9.3), whereas the diagonal-dominance heuristic is a known, explicit assumption of the method. I also note that the paper has independent support: k=1 reproduces the proven Shanks asymptotic, and the leading-order Theorem 5 matches the random-matrix calculation, so the central claim is plausible. The concern is the missing analyticity bridge to the derivative results, not the overall heuristic framework.","tokens_in":1083,"tokens_out":834,"duration_ms":277960,"concrete_test":"For k=2, take (8.1) and compute the Laurent expansion of A'(0)+W_1+W_2 in α and β up to principal parts, using the Laurent expansions ζ(1+x)=1/x+O(1), ζ'(1+x)/ζ(1+x)=-1/x+O(1), the explicit form of A_{α,β}(δ) in (9.3), and its finite-prime truncation. Verify that the residues at α=0, β=0, and α=β vanish identically. If any principal-part coefficient is nonzero, the main term is not holomorphic and the derivative extraction in Section 8 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Remark 5 concedes that holomorphy of the RHS of (1.1) in the shifts is not obvious, and cites Lemma 6.7 of [4] for the permutation sum. But the expression in (1.1) is not the standard CFZ ratio permutation sum: it is the δ-derivative of a sum containing a single denominator zeta, followed by one-swap terms. In the post-δ-derivative form (8.1), the one-swap terms contain factors ζ(1+α_ℓ−α_j)/ζ(1+α_ℓ), which have poles at α_j=α_ℓ, and the zero-swap term has ζ'(1+α_j)/ζ(1+α_j), with poles at α_j=0. The paper does not show that these poles cancel after summing over j. If a residue survives at any α_j=0 or α_j=α_ℓ, then the main term of Conjecture 1 is singular exactly where the conjecture is used, the Taylor expansion in Section 8 is illegitimate, and Conjectures 3–5 do not follow. The cited lemma may concern a different permutation sum; no reduction of (1.1) to that setting is provided.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper formulates Conjecture 1, a Ratios-Conjecture-style asymptotic for the discrete shifted moments of products of Riemann zeta functions evaluated at the non-trivial zeros, assuming RH and small shifts. The main term is presented as the δ-derivative of an integral of a one-denominator ratio Z_{α_1,…,α_k,δ} plus one-swap terms, together with a T/(2π) log(T/(2π)) term and an O(T^{1/2+ε}) error. From this, the authors derive leading-order asymptotics for moments of mixed derivatives (Conjectures 3 and 4), which specialize to a new conjecture for the second moment of ζ′(1/2+iγ) (Conjecture 5) with numerical support. A parallel random-matrix model is developed, giving exact expressions for shifted characteristic-polynomial averages and leading-order derivative moments that match the zeta conjectures.","tokens_in":23409,"tokens_out":12407,"duration_ms":114771,"significance":"If Conjecture 1 holds, the paper provides a unified framework that generalizes the authors' earlier work on Shanks' conjecture to arbitrary mixed derivative moments and yields the first full asymptotic conjecture for the second moment of ζ′(ρ). The paper has several strong points: the k=1 case reproduces the proven asymptotic of [14]; the leading-order terms in Conjectures 3–4 agree with the independently obtained random-matrix results in Theorems 4–5; and the k=2 numerics in Section 9.2 show a small residual after subtracting the full conjectured polynomial, providing a falsifiable check. The derivations contain no fitted constants, and the arithmetic terms are explicitly defined Euler products. The main weaknesses are the unproved diagonal-dominance step in the Ratios recipe and an inadequately supported holomorphy assertion for the main term in the shifts; these are load-bearing for the derivative-moment consequences.","major_comments":[{"comment":"The holomorphy of the right-hand side of (1.1) in the shifts is asserted in Remark 5 with a citation to Lemma 6.7 of [4], but this is not demonstrated for the specific expression here. After the δ-differentiation used in (8.1), the zero-swap term contains ζ′(1+α_j)/ζ(1+α_j) and the one-swap terms contain ζ(1+α_ℓ−α_j)/ζ(1+α_ℓ), so poles occur at α_j=0 and α_j=α_ℓ. Section 8 only verifies cancellation of the leading poles after replacing ζ(1+x) by 1/x and A by 1; no proof is given that the exact sum over j is analytic. Since Conjectures 3–5 are obtained by Taylor expansion about α_j=0 and term-by-term differentiation, a surviving pole would invalidate them. The expression is not the standard CFZ permutation sum, so a direct reduction to Lemma 6.7 of [4] is needed, or the conjecture must be stated with an explicit analytic-continuation caveat.","section":"§8, Remark 5 / Eq. (1.1)"},{"comment":"The diagonal-dominance step — discarding all terms with hm≠n_1⋯n_k after Eq. (7.2) — is the load-bearing heuristic that converts the oscillatory Cauchy integral into the algebraic main term Z_{α,δ} and A_{α}(δ). This is the Ratios Conjecture and is unproved; if off-diagonal contributions do not vanish at the asserted level, the main term of Conjecture 1 changes. The paper is transparent about this in the derivation, but the resulting conjectures (3–5) inherit this assumption and it should be stated explicitly when these results are advertised (e.g., in the abstract) rather than only in the derivation.","section":"§7, Step 4"}],"minor_comments":[{"comment":"The side of the rectangle at Re s = 1−c is called the 'left-hand side'; it should be the right-hand side.","section":"§7"},{"comment":"Typo 'requried' in the proof of Lemma 8; also 'truely' in Section 1 and 'occured' in the Acknowledgements.","section":"§6"},{"comment":"Reference [13] is listed as 'in preparation'; Conjecture 2 and the introduction rely on it. If possible, provide a stable reference or state its availability.","section":"§1"},{"comment":"The numerical evidence would be more compelling with a table of residual magnitudes at selected T, since the figures are qualitative; please also state the size of the imaginary part that is discarded.","section":"§9.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a competent conjectural contribution whose main new claims are plausible within the Ratios-Conjecture framework. The load-bearing issue is the holomorphy assertion: the cited lemma in [4] may not cover the non-standard expression in (1.1)/(8.1). Please ask the authors to supply a direct proof or reduction, or to weaken the derivative-moment statements to depend on an explicit analytic-continuation assumption. The reliance on the unpublished companion paper [13] is a secondary completeness concern, not a blocker."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the take. This paper conjectures a full asymptotic for a shifted product of zeta values at the nontrivial zeros, then differentiates in the shifts to get moments of derivatives. The new thing is Conjecture 5: an explicit polynomial (log terms plus Stieltjes constants and prime products) for the second moment of ζ′(1/2+iγ), which was previously unknown. That's a real step beyond the authors' own earlier complex-moment conjecture and beyond the proven Shanks asymptotics. The random-matrix half is solid: Theorems 3 and 4 are exact and proven via Toeplitz determinants, and Theorem 5 gives the RMT leading order that matches Conjecture 3. The k=1 check reproduces the known [14] asymptotic in integral form, and the numerics for the first million zeros show the polynomial fits well.\n\nThe soft spots are mostly inherent to the Ratios recipe. Step 4 of Section 7 assumes all off-diagonal terms vanish; that's an unproved heuristic, and the paper says so. What I'd want clarified before relying on the derivative conjectures is the holomorphy business. Remark 5 cites Lemma 6.7 of [4] for cancellation of poles, but the expression in (1.1) has gone through a δ-derivative and a one-swap sum, and I don't see a direct reduction to that lemma. The terms in (8.1) visibly have poles at α_j=0 and α_j=α_l. The leading-order cancellation is done in Section 8 via Lemma 2, but for the full expansion in Conjecture 5 you need cancellation of the residue at every order, and that's asserted rather than shown. The numerics in Section 9 are encouraging but qualitative: no error bars, no plot of residual growth, no sensitivity to the 1000-prime cutoff. Minor. The error-term convention is a matter of taste; they flag it.\n\nOverall this is a careful conjectural paper. The central claim is plausible, the execution is mostly clean, and the weaknesses are the genre's usual ones plus one specific spot—the holomorphy of the main term—that deserves expansion. Anyone working on discrete moments of zeta or ratios conjectures will want to read it. I'd cite it as the conjectural source for this moment, and I'd send it to a serious peer reviewer. If the authors can either prove the pole cancellation or state it as a separate conjecture, and add a little quantitative numerics, it would be in good shape.","headline":"A genuinely new conjectural formula for the second moment of ζ′ at its zeros, built from the Ratios recipe; the main risk is unproved pole cancellation in the shift variables.","tokens_in":23994,"tokens_out":2548,"would_cite":true,"duration_ms":28416,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper conjectures that every product of shifted zeta functions at the nontrivial zeros has a full asymptotic expansion whose derivatives in the shifts yield all integer moments of mixed derivatives, including a new explicit formula for","keywords":["Riemann zeta function","discrete moments at zeros","derivatives of zeta","Shanks conjecture","Ratios Conjecture","random matrix theory","characteristic polynomials","Stieltjes constants"],"falsifier":"Compute Σ_{0<γ≤T} ζ′(1/2+iγ)^3 numerically for increasing T: Conjecture 3 predicts the leading term is (1/24)(T/2π)(log(T/2π))^4, so a persistent relative deviation larger than T^{−1/2+ε} at large T would rule out the conjecture. A more direct check is to compute the off-diagonal sums in the recipe's expansion (7.2) for small k and verify that they decay relative to the diagonal terms at the asserted rate.","tokens_in":23008,"feed_emoji":"🔢","tokens_out":8331,"duration_ms":87898,"temperature":0.7,"pith_summary":"This paper conjectures the complete asymptotic expansion of any product of Riemann zeta functions evaluated at the nontrivial zeros, with small shifts added to each argument. Because differentiating with respect to a shift turns ζ(1/2+iγ+α) into ζ^(n)(1/2+iγ), differentiating the conjecture yields the leading-order and full asymptotic expansions for integer moments of mixed derivatives. The authors reach the same conjecture from two independent routes: a random matrix model, where the same combinatorial identities appear for characteristic polynomials, and the ratios recipe, which converts the discrete sum into a Cauchy integral and then into an Euler-product main term. The flagship concrete output is a full asymptotic for the second moment of ζ′(1/2+iγ), with lower-order terms built from Stieltjes constants and products over primes, supported by numerical data up to the millionth zero. The entire structure depends on an unproved heuristic: that off-diagonal terms in the recipe's Dirichlet-series expansion oscillate away and contribute only to the error term.","feed_headline":"Shifted-sum conjecture yields every zeta-derivative moment","feed_subtitle":"Differentiate one shift formula, get every mixed derivative moment, including a full asymptotic for ζ′ squared.","key_machinery":"The engine is a generating integral. Cauchy's theorem turns the discrete sum over zeros into a contour integral of (ζ′/ζ)(s) times the shifted product; the functional equation rewrites it so that the integrand becomes a d/dδ derivative of ζ(s+δ)/ζ(s) times the product. The ratios recipe then replaces ζ(s) by its Dirichlet series and each numerator ζ by its approximate functional equation, keeps only diagonal terms where h m = n_1···n_k, and resums the surviving Euler products into the rational functions Z and the arithmetic factor A. The zero- and one-swap terms are exactly the choices of χ factors that contain equal numbers of χ(s) and χ(1−s), so the oscillatory pieces drop out. Taking d/dδ","core_discovery":"Under the Riemann Hypothesis and mild conditions on shifts α_j, Conjecture 1 states that the discrete shifted moment Σ_{0<γ≤T} ζ(1/2+iγ+α_1)···ζ(1/2+iγ+α_k) equals the δ-derivative at δ=0 of an integral over t of one zero-swap term Z_{α_1,...,α_k,δ} plus k one-swap terms (t/2π)^{−α_j−δ} Z_{...,−δ,...,−α_j}, together with (T/2π)log(T/2π), and an error O(T^{1/2+ε}). The Z functions are products of zeta values at 1+shift divided by ζ(1+α_j), times an Euler-product arithmetic factor A. Expanding the shifts and differentiating recovers Conjecture 3: the leading asymptotic for a product of n_j-th derivatives is (−1)^{Σ n_j + k} n_1!···n_k!/(Σ n_j +1)! times (T/2π)(log(T/2π))^{Σ n_j+1}. The k=2 cas","pith_inferences":["A testable extension the authors do not spell out: applying the same δ-derivative device to ratios with denominator shifts should yield full asymptotics for correlations such as ζ^(m)(1/2+iγ+β)ζ^(n)(1/2+iγ−β), giving joint information about derivatives at nearby zeros.","The leading-order coefficient in Conjecture 3 comes entirely from Vandermonde and complete-homogeneous-symmetric-polynomial identities, independent of the Euler-product factor; this suggests the leading term is more robust than the full expansion, whose arithmetic terms depend on the diagonal-dominance assumption.","Conjecture 5 implies that the real part of Σ ζ′(1/2+iγ)^2 is positive on average with a specific polynomial shape; extending the numerical comparison far beyond the first million zeros would either sharpen confidence in the O(T^{1/2+ε}) error term or reveal its failure.","If the recipe's off-diagonal assumption fails, the random matrix derivation still predicts the leading factorial coefficients, so the two routes would diverge first in the lower-order arithmetic terms rather than in the leading term."],"forward_implications":["If Conjecture 1 holds, Conjecture 3 gives the leading asymptotic for every mixed-derivative moment, and Conjecture 4 gives the kth moment of the nth derivative of ζ at the zeros.","The k=2 case yields Conjecture 5, an explicit full asymptotic for the previously open second moment of ζ′(1/2+iγ), including all lower-order terms.","The k=1 case recovers, in integral form, the full Shanks asymptotic for all higher derivatives of ζ at the zeros, matching an earlier result whose error term becomes power-saving under RH.","The random matrix analogue produces the same factorial coefficient for derivatives of characteristic polynomials evaluated at eigenvalues, strengthening the link between zero statistics and unitary eigenvalue statistics.","Any higher mixed moment can in principle be extracted from Conjecture 1 by Taylor expansion and differentiation, though the paper writes out only the leading term for k≥3."],"supporting_citations":[{"why":"Supplies the Ratios Conjecture recipe used in Section 7 to obtain the main term of Conjecture 1.","marker":"[3]"},{"why":"Provides the integral-moments recipe whose zero- and one-swap diagonalization and error-term convention the paper adopts.","marker":"[2]"},{"why":"Contains Lemma 6.7, which proves the holomorphy of the permutation sum in the shifts, justifying the Taylor expansion and differentiation.","marker":"[4]"},{"why":"Supplies the random matrix model and the exact characteristic-polynomial moment computation used in Sections 3 and 6.","marker":"[16]"},{"why":"Gives the prior discrete mean-value result for derivatives that the k=1 case of Conjecture 1 recovers in integral form.","marker":"[14]"},{"why":"The companion conjecture for complex moments of the first derivative that this paper generalizes to integer mixed-derivative moments.","marker":"[13]"}],"fun_headline_variants":["One shift formula conjectures every zeta-derivative moment","Full asymptotics conjectured for all mixed zeta derivatives","From zero shifts: a conjecture for every zeta moment","New conjecture: all derivatives of ζ at zeros, asymptotically","Random matrix and ratio conjectures unify zeta moment expansions"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The conjecture rests on the recipe's unproved diagonal-dominance step—after substituting the Dirichlet series and approximate functional equations, all terms with h times m not equal to n_1···n_k are assumed to oscillate in t and vanish into the O(T^{1/2+ε}) error—together with the asserted holomorphy of the surviving permutation sum in the shifts.","fun_headline_variants_meta":{"raw":{"variants":["One shift formula conjectures every zeta-derivative moment","Full asymptotics conjectured for all mixed zeta derivatives","From zero shifts: a conjecture for every zeta moment","New conjecture: all derivatives of ζ at zeros, asymptotically","Random matrix and ratio conjectures unify zeta moment expansions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000738,"raw_usage":{"total_tokens":3125,"prompt_tokens":731,"completion_tokens":2394,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":475,"completion_tokens_details":{"reasoning_tokens":2311}},"tokens_in":475,"tokens_out":2394,"duration_ms":18935,"temperature":1.0,"reasoning_tokens":2311,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T21:43:54.414780+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute Σ_{0<γ≤T} ζ′(1/2+iγ)^3 numerically for increasing T: Conjecture 3 predicts the leading term is (1/24)(T/2π)(log(T/2π))^4, so a persistent relative deviation larger than T^{−1/2+ε} at large T would rule out the conjecture. A more direct check is to compute the off-diagonal sums in the recipe's expansion (7.2) for small k and verify that they decay relative to the diagonal terms at the asserted rate.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the integral-moments recipe whose zero- and one-swap diagonalization and error-term convention the paper adopts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains Lemma 6.7, which proves the holomorphy of the permutation sum in the shifts, justifying the Taylor expansion and differentiation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the random matrix model and the exact characteristic-polynomial moment computation used in Sections 3 and 6."},{"cited_title":"Hughes and A","cited_arxiv_id":null,"evidence_quote":"Gives the prior discrete mean-value result for derivatives that the k=1 case of Conjecture 1 recovers in integral form."},{"cited_title":"Hughes and A","cited_arxiv_id":null,"evidence_quote":"The companion conjecture for complex moments of the first derivative that this paper generalizes to integer mixed-derivative moments."}],"review_version":1}