{"id":"4e70b9e8-1e1f-46e2-81ae-badd49511afc","arxiv_id":"2602.06199","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Under RH, the paper derives new explicit bounds for Re(ζ'/ζ), |ζ|, |1/ζ|, and |ζ'/ζ| on Re(s)=1, improving known lower-order constants.","lead":"This paper proves explicit bounds for the real part of the logarithmic derivative of the Riemann zeta function on the line Re(s)=1, assuming the Riemann hypothesis. It uses bandlimited extremal functions to refine the lower-order constants in earlier estimates.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Core proof is internally consistent; the load-bearing risk is the unverified numerical/calculus inequalities determining the final constants, which need independent certification.","rationale":"I read the paper in good faith and followed the main proof chain: Lemma 4, Lemma 5, construction of h±, Lemma 6, Lemma 7, Lemma 8, derivation of Theorem 1, and then the Selberg-formula route to Theorem 3. The structural mathematics is sound and I did not find a fatal error. The improvements advertised—reducing the lower-order error term in |ζ'/ζ| from (log log t)²/log t to log log t/log t—appear genuine, provided the numerical constants are correct. The reader's verdict of CONDITIONAL with high confidence is appropriate. My stress-test diverges from the reader's weakest-assumption statement (RH) and focuses instead on the numerical/calculus assertions, which the reader's rationale also notes as a caveat. These assertions are load-bearing because they set the explicit constants, but they are also concrete and independently checkable. Therefore I recommend no change to the conditional verdict, with the concrete test above serving as the appropriate verification step.","tokens_in":15650,"tokens_out":54796,"duration_ms":440213,"concrete_test":"Write a single interval-arithmetic script (e.g., in Mathematica interval arithmetic or a verified interval library) that certifies all of the following: (1) for πΔ ≥ log 9, the functions Δ ↦ 2e^{πΔ} ε±(Δ) ∓ 8 log(2e^{πΔ}) are decreasing and their endpoint values give η+ = 8.6544 and η− = 6.9856; (2) the trigonometric polynomial in Lemma 12 has minimum ≥ 0 on [0, 2π]; (3) for t ≥ e^18, the bound ((e^{λ0}+1)/(2λ0))(16 log log t − 17.308)/log²t + c/9 + 3.2/t² ≤ 3.648/log t holds. If the script returns a certified yes, the numerical constants stand; if any component fails, the affected theorem needs re-evaluation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 3, and Theorem 1 on which it relies) is mathematically coherent: the Guinand–Weil explicit formula and bandlimited majorants/minorants are applied in a self-consistent way, and I found no internal contradiction or circular step. The weakest point is not the Riemann Hypothesis—it is an explicit, transparent premise—but the set of numerical/calculus assertions that fix the final constants. In §4.1, the bound ε±(Δ) ≤ ±8 log(2e^{πΔ})/(2e^{πΔ}) ∓ η±/(2e^{πΔ}) for πΔ≥log9 is justified only by an unproved monotonicity claim ('It can be shown that the functions ... are decreasing'). In Lemma 12, the extension of an inequality to x≥81 rests on a numerical verification that a trigonometric polynomial has minimum 0. In §7, the error terms are absorbed by the bound ... < 3.648/log t without derivation. Each of these is finite and checkable, but if any is wrong the constants in Theorems 1–3 shift, weakening or altering the advertised improvement over (1.3). This verification gap is the most load-bearing concern for the paper's central numerical claims.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"Assuming the Riemann hypothesis, the paper proves explicit two-sided bounds for ±Re ζ'/ζ(1+it) for t ≥ e^18 (Theorem 1), then uses them to refine the Lamzouri–Li–Soundararajan bounds for |ζ(1+it)| and |1/ζ(1+it)| (Theorem 2) and to improve the Chirre–Valås–Simonič bound for |ζ'/ζ(1+it)| (Theorem 3). The main advertised novelty is that the lower-order error term in the bound for |ζ'/ζ(1+it)| is reduced from (log log t)^2/log t to log log t/log t, with an additional negative 1/log t term. The proofs combine the Guinand–Weil explicit formula with extremal bandlimited majorants/minorants of the Poisson kernel, together with Perron-type identities and estimates adapted from [10].","tokens_in":15990,"tokens_out":5119,"duration_ms":51133,"significance":"If all the numerical and calculus checks are made fully rigorous, this is a meaningful contribution to explicit conditional estimates on the edge of the critical strip. The proofs are structurally detailed, the Riemann hypothesis is an explicit and transparent premise rather than a hidden assumption, and the application of the Beurling–Selberg extremal machinery is coherent. The improvement over (1.3) is conceptually interesting and the refinements of (1.1)–(1.2) are useful. However, several load-bearing constants are justified only by statements such as 'numerically one can verify' or 'it can be shown', without reproducible code or complete analytic verification. Since these constants determine the advertised improvements, the paper is not yet in a fully acceptable state.","major_comments":[{"comment":"The constants η+ = 8.6544 and η− = 6.9856 in (4.3), and hence Theorems 1–3, depend on the assertion that the functions 2e^{πΔ}ε±(Δ)∓8 log(2e^{πΔ}) are decreasing for πΔ ≥ log 9. This is stated only in the footnote ('It can be shown') with no proof. This is a load-bearing calculus check; please provide a complete proof or certified interval-arithmetic code.","section":"§4.1 / footnote 3"},{"comment":"The extension of (5.1) from x ≥ 100 to x ≥ 81 is essential for the lower bound (1.5). For p = 2 in the range 81 ≤ x ≤ 100, the argument reduces the problem to a trigonometric polynomial whose minimum is asserted to be 0 'by numerical verification', with no code or rigorous error bound. Similarly, the later claim that the supremum over 81 ≤ x ≤ (73.2)^2 of the displayed expression is at most 0.249 is stated only as 'Numerically'. Since the constant 0.249 appears in the final bound (1.5), these finite verifications must be made reproducible and rigorous.","section":"Lemma 12"},{"comment":"The final absorption step leading to Theorem 3, namely the assertion that the combination of error terms is < 3.648/log t for t ≥ e^18, is stated without derivation. This step is load-bearing because the advertised improvement over (1.3) depends on the resulting coefficient 9.0581 and the negative 4.7/log t term. Please provide the explicit estimate, including how the 13652/log^6 t term, the 17.308/log^2 t term, and the 3.2/t^2 term are dominated by 3.648/log t in the stated range.","section":"§7, after (7.6)"}],"minor_comments":[{"comment":"The theorem states the coefficient of log log t/log t as 2.6, while the proof of (1.4) concludes with 2.676. Since 2.6 gives a weaker bound than 2.676, the theorem is valid but the mismatch is confusing; please harmonize the display and the proof.","section":"Theorem 2 / §6.1"},{"comment":"The theorem states −4.7/log t, while the proof obtains −4.773/log t before 'This implies the desired result'. Again, the stated constant is weaker, but the discrepancy should be noted or the stronger constant inserted.","section":"Theorem 3 / §7"},{"comment":"The bound '2(e^{πΔ}+e^{-πΔ})/c±_Δ ≤ 3.325...' is asserted without derivation. This is a minor issue, but since the paper aims at explicit constants, a one-line verification would be helpful.","section":"Lemma 8"},{"comment":"The paper uses 'numerically one can verify' in several places without indicating whether the verification is fully rigorous or merely floating-point. For an explicit-constants paper, the authors should either supply a proof, include reproducible code, or clearly mark the status of each numerical check.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The reader's report and stress-test note match my own reading: the core argument is coherent and not circular, and the main obstacle is the certification of several finite numerical/calculus assertions that determine the final constants. I would advise requesting full proofs or certified code for the items in §4.1, Lemma 12, and §7 before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing you should know: this is a real improvement of explicit conditional bounds, and the core argument is sound. The new bit is Theorem 3's error term: under RH they bring the lower-order term for |ζ'/ζ| down from (log log t)^2/log t to log log t/log t, plus a negative 1/log t term. Theorem 2 also refines Lamzouri–Li–Soundararajan with better 1/log log t constants and a negative lower-order term. That is a genuine contribution, not a repackaging.\n\nWhat is good: the proof strategy is natural—Guinand–Weil with bandlimited majorants/minorants for the Poisson kernel—and they do a genuinely more careful job on the Gamma term than earlier work. Lemmas 6–8 are the technical core and look right. The structure is transparent. The self-citations are appropriate; they are directly extending prior results in the same line.\n\nThe soft spots: a cluster of numerical/calculus assertions that fix the final constants. Footnote 3 says the relevant functions are decreasing for πΔ≥log 9 with no proof. Lemma 12's extension from x≥100 to x≥81 rests on \"one can verify numerically\" that a trig polynomial has minimum 0. And at the end of Section 7 the < 3.648/log t absorption is asserted without derivation. Each of these is finite and checkable, but they are load-bearing: if any is off, the constants shift and the advertised improvement over (1.3) weakens. I don't see any circularity; the RH is an explicit premise. The error-term bookkeeping is otherwise careful. There is no shipped code, so the numeric checks are not independently reproducible. That is the main thing a referee should push on.\n\nThe paper deserves a serious referee. It is a conditional explicit-bounds paper with real, if incremental, gains, and the method is coherent. The reader's Pith Report is about right: significance around 6, novelty around 7, soundness around 7 with the numerical-verification caveat. If it lands in your inbox, send it to review and ask for either full analytic proofs of those inequalities or reproducible verification.","headline":"Solid conditional improvement; the advertised constants depend on unverified numerical inequalities that a referee should pin down.","tokens_in":16412,"tokens_out":1334,"would_cite":true,"duration_ms":14537,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","41A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Assuming the Riemann hypothesis, this paper proves explicit bounds for ζ and its logarithmic derivative on Re s = 1 with uniform error terms valid for all t ≥ e^18, improving the lower-order error from (log log t)^2/log t to log log t/log t","keywords":["Riemann zeta-function","Riemann hypothesis","bandlimited majorants","logarithmic derivative","explicit bounds","critical strip","Poisson kernel","extremal functions"],"falsifier":"Compute |ζ'/ζ(1+it)| to rigorous precision at any t ≥ e^18 (for instance t = 10^k for k = 8, 9, ...). If any value exceeds 2 log log t − 0.4989 + 9.0581 log log t/log t − 4.7/log t, Theorem 3 is false. More fundamentally, locating a non-trivial zero off the line Re s = 1/2 would falsify the hypothesis under which all bounds are proved.","tokens_in":15592,"feed_emoji":"📈","tokens_out":12269,"duration_ms":110415,"temperature":0.7,"pith_summary":"Assuming the Riemann hypothesis, this paper gives explicit, uniform bounds for ζ and its logarithmic derivative on the line Re s = 1, valid for every t ≥ e^18 (about 6.57 × 10^7). The central improvement is in the lower-order terms: |ζ'/ζ(1+it)| is shown to be ≤ 2 log log t − 0.4989 + 9.0581 log log t/log t − 4.7/log t, replacing the previous (log log t)^2/log t error by a term one log-factor smaller, and in particular ≤ 2 log log t for t ≥ 10^30. For |ζ(1+it)| and 1/|ζ(1+it)|, the paper refines the century-old asymptotic estimates by improving the constants inside the 1/log log t corrections and introducing negative lower-order terms. All bounds are conditional: if the Riemann hypothesis fails, they do not apply.","feed_headline":"Error term for ζ'/ζ(1+it) cut from (log log t)^2 to log log t","feed_subtitle":"Assuming RH, new explicit bounds valid for all t ≥ e^18 sharpen the classic estimates for zeta near the 1-line.","key_machinery":"The central mechanism is the Poisson kernel h(x) = 1/(2(1/4 + x^2)), which enters through the identity Re ζ'/ζ(1+it) = Σ_γ h(t−γ) − 1/2 log(t/2π) + O(1/t^2). Because h is not bandlimited, the paper replaces it by extremal bandlimited majorants and minorants h^±_∆: entire functions of exponential type 2π∆ that lie above (resp. below) h and are optimal in the one-sided bandlimited approximation sense, with Fourier transforms supported on [−∆,∆]. Substituting h^±_∆ into the Guinand–Weil explicit formula converts the zero sum into gamma-function, prime-power, and boundary terms; a refined Stirling expansion of the gamma term retains lower-order corrections, and optimizing the bandwidth by taking","core_discovery":"The paper establishes explicit two-sided bounds for Re ζ'/ζ(1+it) under the Riemann hypothesis, for all t ≥ e^18. As consequences it obtains: |ζ'/ζ(1+it)| ≤ 2 log log t + 0.0784 − γ + 9.0581 log log t/log t − 4.7/log t (so that in particular |ζ'/ζ(1+it)| ≤ 2 log log t for t ≥ 10^30); |ζ(1+it)| ≤ 2e^γ( log log t − log 2 + 1/2 + 0.2674/log log t − 2.676 log log t/log t ); and 1/|ζ(1+it)| ≤ (12e^γ/π^2)( log log t − log 2 + 1/2 + 5/(8 log log t) + 10.7084/(log log t)^2 ). These are the sharpest explicit conditional bounds currently known on the 1-line, with optimized constants in all lower-order terms.","pith_inferences":["The same bandlimited-majorant-plus-refined-Stirling recipe should transfer to other L-function families, yielding the same one-log-factor reduction in error terms for their logarithmic derivatives on the edge of the strip.","The bounds can be stress-tested numerically: a rigorous high-precision evaluation of |ζ'/ζ(1+it)| at some t ≥ e^18 that violates the stated inequality would not refute RH but would point to a specific flaw in the proof, while agreement would support the claimed shape of the error term.","The method could be pushed further by keeping even more terms in the Stirling expansion, which would likely lower the constants 9.0581 and 4.7 and extend the range of t for which the bound dips below 2 log log t.","A natural next target is the derivative (ζ'/ζ)' on the same line, using the same extremal functions; the refined gamma term developed here would enter directly into its explicit formula."],"forward_implications":["If the theorem is correct, the conditional order of magnitude of ζ'/ζ on the 1-line is settled: for t ≥ 10^30 it is at most 2 log log t, matching the expected asymptotic and improving the previous explicit range.","The one-sided bounds on Re ζ'/ζ(1+it) directly imply refined estimates for |ζ(1+it)| and 1/|ζ(1+it)|, valid uniformly for all t ≥ e^18, with smaller 1/log log t constants and new negative lower-order corrections.","The error term in the log-derivative bound is reduced by a full log factor: from (log log t)^2/log t to log log t/log t, making the bound numerically meaningful at substantially smaller heights.","The optimized choice e^{π∆} = log t / 2 shows the leading 2 log log t is the only thing that sets the scale; the error terms come from the Stirling expansion and prime sums rather than from a rebalancing of the bandwidth."],"fun_headline_variants":["Zeta log derivative error cut to log log t under RH","Explicit bounds for ζ(1+it) sharpened under RH","New explicit RH bounds tighten ζ and ζ' near 1-line","RH yields explicit ζ(1+it) bounds for every t≥e^18","Littlewood ζ(1+it) estimates refined with optimized constants"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The Riemann Hypothesis: all main theorems assume every non-trivial zero of ζ(s) has real part 1/2; if a single non-trivial zero lies off that line, the bounds have no standing without substantial modification.","fun_headline_variants_meta":{"raw":{"variants":["Zeta log derivative error cut to log log t under RH","Explicit bounds for ζ(1+it) sharpened under RH","New explicit RH bounds tighten ζ and ζ' near 1-line","RH yields explicit ζ(1+it) bounds for every t≥e^18","Littlewood ζ(1+it) estimates refined with optimized constants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00113,"raw_usage":{"total_tokens":4554,"prompt_tokens":783,"completion_tokens":3771,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":3686}},"tokens_in":527,"tokens_out":3771,"duration_ms":27589,"temperature":1.0,"reasoning_tokens":3686,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T03:57:18.458203+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute |ζ'/ζ(1+it)| to rigorous precision at any t ≥ e^18 (for instance t = 10^k for k = 8, 9, ...). If any value exceeds 2 log log t − 0.4989 + 9.0581 log log t/log t − 4.7/log t, Theorem 3 is false. More fundamentally, locating a non-trivial zero off the line Re s = 1/2 would falsify the hypothesis under which all bounds are proved.","supporting_citations":[],"review_version":1}