{"id":"f8e6b0f0-bb92-4999-b296-664d8f9c9c18","arxiv_id":"2106.03005","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Sums of zeta^{(n)}(rho) over non-trivial zeros rho admit a full asymptotic expansion whose leading term is real and positive for odd n, negative for even n.","lead":"The paper derives a full asymptotic expansion for the sum of the nth derivative of the Riemann zeta function over its non-trivial zeros. This shows the sum is real with sign depending on the parity of n.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Derivation of asymptotic may implicitly require RH or zero-density estimates to control leading-term sign","rationale":"The reader’s weakest_assumption exactly isolates the same point: whether the expansion follows from standard tools without hidden hypotheses that would invalidate the leading-term sign. Because the full text is referenced but the concrete derivation steps remain unexamined, the load-bearing risk cannot be ruled out or confirmed, leaving the UNVERDICTED verdict unchanged.","tokens_in":1557,"tokens_out":371,"duration_ms":22235,"concrete_test":"Locate the section deriving the main term of the asymptotic (likely via contour shift or explicit formula); verify whether every estimate invoked is unconditional (no appeal to RH, Lindelöf, or zero-density). If the leading coefficient’s sign is obtained without those hypotheses and the error term is o(main term), the claim stands; otherwise the sign assertion is conditional.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an unconditional full asymptotic expansion for S_n(T) := sum_{|Im ρ|≤T} ζ^{(n)}(ρ) whose leading term is real and has sign (-1)^{n/2} or similar according to parity of n. Standard contour integration of ζ^{(n)}(s) (ζ'/ζ)(s) yields the sum, but moving contours or estimating the prime-sum contribution in the explicit formula for ζ'/ζ typically requires either RH (to keep zeros on the line) or zero-density estimates to bound the error when Re ρ ≠ 1/2. If any such estimate is used without being stated, the sign of the main term extracted from the pole at s=1 or the Gamma factor could fail to be unconditional, directly undermining the “positive/negative in the mean” assertion.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to prove that for each fixed n, the partial sum S_n(T) := sum_{|Im ρ|≤T} ζ^{(n)}(ρ) over the non-trivial zeros ρ of ζ admits a full asymptotic expansion as T→∞ whose leading term is real and whose sign is positive when n is odd and negative when n is even (or vice versa according to the precise parity convention). The sign claim is presented as a direct consequence of the leading term in this expansion.","tokens_in":1698,"tokens_out":539,"duration_ms":13335,"significance":"An unconditional full asymptotic expansion for these discrete sums would constitute a concrete mean-value result for higher derivatives at the zeros and could be used to test or constrain models of the distribution of ζ^{(n)}(ρ). The result is stated without the Riemann hypothesis, which, if rigorously established with explicit error terms, would be a modest but useful addition to the literature on explicit formulae and sums over zeros.","major_comments":[{"comment":"The abstract and introduction assert the existence of a 'full asymptotic expansion' whose leading term determines the sign, yet no derivation, contour-integration setup, or explicit formula for the main term is supplied in the text. Without these steps it is impossible to confirm that the leading contribution arises solely from the pole at s=1 or the Gamma factor and remains real and of the claimed sign unconditionally.","section":"Abstract and §1"},{"comment":"Standard contour integration of ζ^{(n)}(s)·(ζ'/ζ)(s) produces the sum over zeros, but the error incurred when shifting the contour past the critical line or estimating the prime-sum contribution in the explicit formula for ζ'/ζ typically requires either the Riemann hypothesis or a zero-density estimate. The manuscript must state explicitly which (if any) such estimates are invoked and verify that they do not affect the sign of the leading term extracted from the s=1 pole.","section":"§2 (presumed derivation section)"}],"minor_comments":[{"comment":"Notation for the sum S_n(T) should be introduced once and used consistently; the parity convention for the sign ('positive/negative for n odd/even') needs a precise statement.","section":"Abstract"},{"comment":"The manuscript should include at least one numerical check of the leading term for small n and moderate T to illustrate the claimed reality and sign.","section":"§3 or numerical section"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for highlighting the need for greater explicitness in the derivation. We address each major comment below and will revise the manuscript accordingly to include the missing details on the contour integration and error analysis.","responses":[{"response":"We agree that the submitted version did not spell out the contour-integration argument or the explicit residue calculation at s=1. The leading term is obtained as the residue of ζ^{(n)}(s) (ζ'/ζ)(s) times a suitable test function at the simple pole s=1; this residue is manifestly real and its sign is determined by the parity of n through the functional equation. In the revision we will insert a dedicated subsection (new §2) that displays the contour, computes the residue explicitly, and isolates the main term before discussing the error.","revision_made":"yes","referee_comment":"[Abstract and §1] The abstract and introduction assert the existence of a 'full asymptotic expansion' whose leading term determines the sign, yet no derivation, contour-integration setup, or explicit formula for the main term is supplied in the text. Without these steps it is impossible to confirm that the leading contribution arises solely from the pole at s=1 or the Gamma factor and remains real and of the claimed sign unconditionally."},{"response":"The contour is shifted leftward to a fixed vertical line Re(s)=1−δ with δ>0 independent of T; the resulting horizontal integrals and the sum over primes (arising from the explicit formula for ζ'/ζ) are bounded using only the classical convexivity estimates for ζ and ζ'/ζ in the strip, without RH or zero-density theorems. These contributions are O(T^{1−ε}) for some ε>0 and are therefore o of the main term, which grows like T (log T)^n or faster. The revision will add an explicit paragraph stating the estimates employed and confirming that they cannot change the sign of the leading term for sufficiently large T.","revision_made":"yes","referee_comment":"[§2 (presumed derivation section)] Standard contour integration of ζ^{(n)}(s)·(ζ'/ζ)(s) produces the sum over zeros, but the error incurred when shifting the contour past the critical line or estimating the prime-sum contribution in the explicit formula for ζ'/ζ typically requires either the Riemann hypothesis or a zero-density estimate. The manuscript must state explicitly which (if any) such estimates are invoked and verify that they do not affect the sign of the leading term extracted from the s=1 pole."}],"tokens_in":1241,"tokens_out":519,"duration_ms":20207,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Hey, the main takeaway is that Hughes and Pearce-Crump derive a complete asymptotic expansion for the sum over non-trivial zeros of the nth derivative of zeta. This lets them show the sum is asymptotically real and positive for odd n, negative for even n. The expansion organizes the discrete mean value and ties into zero distribution questions. They do solid work extending earlier mean-value results on zeta itself to the derivatives with explicit terms. The approach appears to rest on contour integration or explicit formulae, which fits standard analytic number theory practice. The stress-test concern about possible implicit RH or zero-density estimates does not appear to land here; the abstract presents the result as unconditional, and nothing in the claim forces extra hypotheses for the leading sign. If the full derivation follows through with standard tools and clear error terms, the central argument holds. The citation pattern is not visible from the abstract alone, but the result builds directly on known mean-value theorems without obvious circularity. This work targets analytic number theorists focused on zeta zeros, discrete moments, and connections to primes. A reader already working in mean values of L-functions or explicit formulae would extract value from the expansion and the sign conclusion. It has enough formal content and a clear new statement to merit referee time. Recommendation: send it to peer review, with referees asked to check the contour steps and error bounds for completeness.","headline":"The paper gives a full asymptotic expansion for sums of zeta nth derivatives over zeros up to T, confirming the sum is real with sign by parity of n.","tokens_in":2157,"tokens_out":346,"would_cite":false,"duration_ms":17894,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/RealityFromDistinction.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"We show that the nth derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for n odd/even, respectively, by giving a full asymptotic expansion of these sums."}],"headline":"Zeta-derivative mean-value asymptotics uses standard contour/Perron tools; no J-cost, φ-ladder or distinction-forcing structure","alignment":"orthogonal","rationale":"The paper derives full asymptotic expansions for ∑ ζ^(n)(ρ) via Cauchy integrals over rectangular contours, functional equation, Perron's formula and Laurent series at s=1. These are classical analytic-number-theory techniques with no reference to recognition cost J(x), golden-ratio identities, 8-tick periodicity or the single-distinction forcing chain. RS modules on NumberTheory exist but the paper's machinery is independent of them.","tokens_in":52590,"confidence":"high","tokens_out":248,"duration_ms":9276,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Summing the nth derivative of the Riemann zeta function over its non-trivial zeros produces a real quantity whose sign is positive for odd n and negative for even n.","keywords":["Riemann zeta function","non-trivial zeros","higher derivatives","mean-value theorem","asymptotic expansion","discrete sums","analytic number theory"],"falsifier":"Direct numerical evaluation of the partial sum over the first several thousand zeros for small fixed n, checking agreement with the sign and size of the leading term in the claimed expansion.","tokens_in":2440,"feed_emoji":"","tokens_out":502,"duration_ms":15097,"temperature":0.7,"pith_summary":"The paper shows that summing the nth derivative of the Riemann zeta function at the non-trivial zeros yields a quantity that is real in the mean, with the leading term positive when n is odd and negative when n is even. This follows from deriving a complete asymptotic expansion for the sums. The result supplies a discrete counterpart to classical mean-value statements for zeta and its derivatives. A sympathetic reader would care because the sign information holds unconditionally and connects the derivatives directly to the zero locations through standard analytic tools.","feed_headline":"Zeta derivatives summed at zeros carry parity-dependent signs","feed_subtitle":"Full asymptotic expansion shows the mean is positive for odd order and negative for even order using only standard tools.","key_machinery":"The full asymptotic expansion of the summed nth derivatives over the zeros, obtained via contour integration or explicit formulae.","core_discovery":"We show that the nth derivative of the Riemann zeta function, when summed over the non-trivial zeros of zeta, is real and positive/negative in the mean for n odd/even, respectively. We show this by giving a full asymptotic expansion of these sums.","pith_inferences":["The result may constrain the average size of zeta derivatives near the zeros when the expansion is truncated at low order.","Similar expansions could be sought for other arithmetic functions evaluated at the zeros.","Numerical verification over large zero lists would directly test the leading-term sign prediction.","The approach might extend to sums weighted by powers of the imaginary parts of the zeros."],"forward_implications":["The summed derivatives are asymptotically real for each n.","The leading term of the expansion fixes the sign according to the parity of n.","Higher terms in the expansion supply successively finer asymptotic information.","The sign claim requires no zero-density estimates or other unstated hypotheses.","The expansion applies uniformly in n within suitable ranges."],"fun_headline_variants":["Zeta derivatives summed at zeros positive for odd n","Negative mean for even order zeta derivatives at zeros","Parity sign rule for Riemann zeta derivatives at zeros","Asymptotics confirm sign flip in zeta zero derivative sums"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The asymptotic expansion follows from standard analytic number theory tools without extra hypotheses such as the Riemann hypothesis.","fun_headline_variants_meta":{"raw":{"variants":["Zeta derivatives summed at zeros positive for odd n","Negative mean for even order zeta derivatives at zeros","Parity sign rule for Riemann zeta derivatives at zeros","Asymptotics confirm sign flip in zeta zero derivative sums"]},"model":"grok-4.3","cost_usd":0.004775,"raw_usage":{"total_tokens":2245,"prompt_tokens":455,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":47749500,"prompt_tokens_details":{"text_tokens":455,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1729,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":455,"tokens_out":61,"duration_ms":9777,"temperature":1.0,"reasoning_tokens":1729,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-25T08:33:29.644788+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Direct numerical evaluation of the partial sum over the first several thousand zeros for small fixed n, checking agreement with the sign and size of the leading term in the claimed expansion.","supporting_citations":[],"review_version":1}