{"id":"056520de-14b4-46df-bf2f-726aba565d6f","arxiv_id":"2605.27552","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":2.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Riemann supplied explicit values for the derivatives of log ζ(s) at s=1/2, including ζ'(1/2)/ζ(1/2) = π/4 + γ/2 + log(8π)/2 and a second-derivative formula 8 - π²/4 - 2G + 2 ∑ 1/α_n².","lead":"This paper presents explicit formulas for the first and second logarithmic derivatives of the Riemann zeta function at s=1/2 drawn from Riemann's posthumous notes. A smart generalist might read it to see concrete historical expressions involving standard constants like π, γ, and Catalan's constant G in the context of analytic number theory.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Attribution of the two explicit formulas to Riemann's Göttingen manuscript rests only on assertion without excerpt or page reference.","rationale":"The reader's weakest_assumption already isolates the precise point of vulnerability—the unverified extraction from the Göttingen manuscript—and the paper's extreme brevity supplies no independent evidence that would mitigate it. No other technical or logical gap in the argument is visible once the historical claim is set aside.","tokens_in":1734,"tokens_out":326,"duration_ms":25434,"concrete_test":"Request the relevant Riemann Nachlass volume from the Göttingen archive (or its digitized surrogate), locate any passage treating log ζ or its derivatives near s = 1/2, and compare the manuscript expressions term-by-term with the two displayed formulas; a mismatch in any constant or in the definition of the α_n sequence falsifies the attribution.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the displayed identities for (log ζ)'(1/2) and the combination of (log ζ)''(1/2) are values that Riemann himself recorded in one of his posthumous papers. The manuscript text supplies only the bare statement that Riemann considered these derivatives and gave explicit values; it contains neither a quotation from the Göttingen document, a citation to a specific shelf-mark or page, nor any derivation showing how the constant terms (π/4, γ, log(8π), G, the sum over α_n) follow from Riemann's own notation or calculations. Consequently the historical fidelity of the formulas cannot be checked from the paper itself.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript asserts that in one of his posthumous papers conserved in Göttingen, Riemann computed and recorded explicit values for the first and second logarithmic derivatives of ζ(s) at s = 1/2. It displays the two formulas ζ'(1/2)/ζ(1/2) = π/4 + γ/2 + log(8π)/2 and ζ''(1/2)/ζ(1/2) − (ζ'(1/2)/ζ(1/2))² = 8 − π²/4 − 2G + 2 ∑_{n=1}^∞ 1/α_n², states that these were shared with mathematicians around 2010, and presents the note as an explanation of those formulas.","tokens_in":1854,"tokens_out":445,"duration_ms":39232,"significance":"If the attribution to Riemann's manuscript is accurate and the formulas are faithfully transcribed, the note would record a previously undocumented calculation by Riemann on the zeta function at the critical line, of potential interest to historians of analytic number theory. The formulas combine classical constants (π, γ, Catalan's G) with a sum that appears to involve the ordinates of zeta zeros, but the manuscript supplies neither a derivation nor any verification that these expressions match Riemann's own notation or calculations.","major_comments":[{"comment":"Abstract, first paragraph: the central historical claim—that the displayed formulas appear in Riemann's Göttingen manuscript—is advanced solely by assertion, with no quotation, shelf-mark, page reference, or excerpt from the document. This absence makes the attribution impossible to verify and is load-bearing for the paper's thesis.","section":"Abstract, first paragraph"},{"comment":"The second displayed formula invokes the sum ∑ 1/α_n² without defining α_n or indicating how the constant terms (8, −π²/4, −2G) arise from Riemann's calculations; no derivation or cross-reference to the manuscript is supplied to support these explicit values.","section":"Abstract"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive comments on our short note. We address each major comment below and indicate planned revisions.","responses":[{"response":"The note is a brief record of formulas shared from the Göttingen collection rather than a critical edition or full transcription. The attribution rests on the author's examination of the relevant posthumous papers. We will revise the abstract and main text to include the specific shelf-mark and page reference for the document in question.","revision_made":"yes","referee_comment":"[Abstract, first paragraph] Abstract, first paragraph: the central historical claim—that the displayed formulas appear in Riemann's Göttingen manuscript—is advanced solely by assertion, with no quotation, shelf-mark, page reference, or excerpt from the document. This absence makes the attribution impossible to verify and is load-bearing for the paper's thesis."},{"response":"The purpose of the note is to present the explicit values recorded by Riemann, not to re-derive them. We will add a parenthetical definition of α_n as the positive ordinates of the non-trivial zeros. The constants are those appearing directly in the manuscript entry; a cross-reference to the relevant page will be supplied in revision. No derivation is provided because none appears in the source document itself.","revision_made":"partial","referee_comment":"[Abstract] The second displayed formula invokes the sum ∑ 1/α_n² without defining α_n or indicating how the constant terms (8, −π²/4, −2G) arise from Riemann's calculations; no derivation or cross-reference to the manuscript is supplied to support these explicit values."}],"tokens_in":1414,"tokens_out":359,"duration_ms":40334,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this is a brief note that writes down two closed-form expressions for the first and second logarithmic derivatives of zeta at s=1/2, attributing them directly to one of Riemann's posthumous papers held in Göttingen. The formulas are given with the usual constants (gamma, log(8 pi), Catalan's G, and the sum over 1/alpha_n squared) and the author mentions having shared the second one informally around 2010 before deciding to publish for reference purposes.\n\nThe paper does what it sets out to do by putting the expressions into print in a compact, readable way. Anyone who has needed these particular values now has a citable source instead of hunting through old notes or computing them from scratch.\n\nThe soft spot is exactly the one the stress-test flags: the attribution is stated as fact but the text contains no quotation from the manuscript, no shelf-mark, no page number, and no indication of how the constants were extracted from Riemann's notation. For a history-of-mathematics note that is a noticeable gap; readers cannot verify the claim from the paper itself. There is also no derivation or numerical check supplied, which keeps the work purely expository.\n\nThis is aimed at analytic number theorists who work with explicit values on the critical line or who track historical expressions for the zeta function. It has too little original content or analysis to justify sending it to referees. I would not bring it to a reading group or cite it in new work, and I would not recommend peer review.","headline":"Short note recording two explicit formulas for log zeta derivatives at 1/2, credited to Riemann but without any source excerpt or page reference.","tokens_in":2337,"tokens_out":384,"would_cite":false,"duration_ms":42251,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Riemann derived explicit formulas for the logarithmic derivatives of zeta at s=1/2 from his unpublished notes.","keywords":["Riemann zeta function","logarithmic derivatives","posthumous papers","critical line","Catalan's constant","Euler-Mascheroni constant","non-trivial zeros"],"falsifier":"Direct inspection of the relevant page in Riemann's conserved notebook to check whether the two displayed identities appear in that form.","tokens_in":2604,"feed_emoji":"📜","tokens_out":683,"duration_ms":28289,"temperature":0.7,"pith_summary":"The paper extracts and presents two formulas that Riemann wrote down for the first and second derivatives of log zeta at the point s=1/2. The first gives zeta prime over zeta at 1/2 as a combination of pi, the Euler-Mascheroni constant, and the log of 8 pi. The second expresses the combination of second derivative over zeta minus the square of the first in terms of 8, pi squared, Catalan's constant, and a sum of reciprocal squares over the ordinates of the non-trivial zeros. A reader would care because these supply concrete numerical values that tie the zeta function's local behavior on the critical line to global constants and the zero locations.","feed_headline":"Riemann supplied explicit log derivatives of zeta at s=1/2","feed_subtitle":"Formulas give first ratio with pi and gamma, second with Catalan's constant plus sum over zero squares","key_machinery":"The logarithmic derivatives of zeta at the fixed point s=1/2, written as explicit combinations of elementary constants and a sum over the squares of the imaginary parts of the zeros.","core_discovery":"In his posthumous papers Riemann considered the derivatives of log zeta(s) at s=1/2 and supplied the explicit evaluations zeta'(1/2)/zeta(1/2) equals pi/4 plus gamma/2 plus log(8 pi)/2 together with zeta''(1/2)/zeta(1/2) minus the square of the first derivative ratio equals 8 minus pi squared over 4 minus 2 G plus 2 times the sum over n of 1 over alpha_n squared.","pith_inferences":["If the sum over 1/alpha_n squared converges rapidly, the second formula could be turned into a practical numerical check against independently computed values of the second derivative.","The appearance of Catalan's constant suggests a possible link between the zeta derivatives at 1/2 and certain alternating series that arise in other contexts of analytic number theory."],"forward_implications":["The first formula supplies a closed numerical value for the logarithmic derivative that can be used in series expansions around the critical line.","The second formula relates the curvature of log zeta at 1/2 to the sum of 1/alpha_n squared, thereby connecting a local analytic quantity to the distribution of zeros.","Both expressions can be inserted into functional equations or product representations without needing to evaluate zeta numerically at that point."],"fun_headline_variants":["Riemann explicit log derivatives of zeta at s=1/2","Log zeta derivatives at 1/2 explicit from Riemann","Riemann's explicit values for zeta log derivatives at 1/2","Explicit log derivatives of zeta at s=1/2 by Riemann","Riemann gives explicit log zeta derivs at s=1/2"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The displayed formulas match what Riemann actually wrote in the manuscript kept in Göttingen.","fun_headline_variants_meta":{"raw":{"variants":["Riemann explicit log derivatives of zeta at s=1/2","Log zeta derivatives at 1/2 explicit from Riemann","Riemann's explicit values for zeta log derivatives at 1/2","Explicit log derivatives of zeta at s=1/2 by Riemann","Riemann gives explicit log zeta derivs at s=1/2"]},"model":"grok-4.3","cost_usd":0.005525,"raw_usage":{"total_tokens":2554,"prompt_tokens":634,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":55253000,"prompt_tokens_details":{"text_tokens":634,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1839,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":634,"tokens_out":81,"duration_ms":22109,"temperature":1.0,"reasoning_tokens":1839,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T14:02:14.495135+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Direct inspection of the relevant page in Riemann's conserved notebook to check whether the two displayed identities appear in that form.","supporting_citations":[],"review_version":1}