{"id":"ea59510c-808e-42da-b32d-2d7498c76d4d","arxiv_id":"2412.02068","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper claims N(σ,T) ≤ K T^{4σ(1-σ)} (log T)^{5-2σ}, but its own proof establishes only the weaker exponent 4.","lead":"This paper derives an explicit zero density estimate for the Riemann zeta function, a bound on how many zeros lie to the right of a vertical line. The proof, however, only yields the classic logarithm power 4, not the improved power 5-2σ promised in the abstract and theorem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 advertises (log T)^{5-2σ}, but the proof terminates with (log T)^4; the claimed improvement over Carlson is not derived.","rationale":"I read the paper's aim as proving an explicit Carlson-type estimate with the improved logarithm exponent 5−2σ. The reader's strongest_claim is exactly right: Theorem 1.1 and the abstract state this exponent, but the proof terminates at log^4 T. This is load-bearing because the entire claimed improvement over the classical O(T^{4σ(1−σ)} log^4 T) rests on that exponent. The proof is otherwise coherent in outline: the mollifier second-moment estimate (4.6) is the kind of input Carlson's method needs, and the constants are made explicit. However, no amount of constant optimization can turn the final log^4 into log^{5−2σ}; it would require an extra T-dependent logarithmic saving of (log T)^{2σ−1} that is not present in the displayed inequalities. I therefore agree with the reader's rejection of the current version. The paper is probably repairable by restating the theorem with log^4 T and recomputing the comparison and table constants, but as submitted the advertised central claim fails. I set verdict_should_be to UNCHANGED because my stress-test reinforces the reader's verdict rather than moving it. I mark agreement_with_reader as partial because the reader's weakest_assumption focused on the imported KLN lemmas, whereas the decisive flaw is the missing exponent; the KLN-lemma concern is real but secondary.","tokens_in":8530,"tokens_out":5852,"duration_ms":55365,"concrete_test":"Re-derive the final estimate by substituting (5.1) with α = σ − 1/log T into (2.4), keeping the prefactor (σ−α)^{-1} = log T explicit. The first term is (C3(σ,T0)/(2π)) T^{4σ(1−σ)} log^4 T. Then search §5 for any inequality that absorbs a factor (log T)^{2σ−1}; if none exists, Theorem 1.1's (log T)^{5−2σ} is not proved. A useful numerical cross-check: for T0 = 3·10^{12} and σ = 0.6, compare the RHS of (1.7) with exponent 5−2σ = 3.8 against the proved log^4 exponent; since log T > 1, the proved RHS is larger for the same K, confirming the theorem is strictly stronger than anything shown.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive issue is at the end of §5. After substituting (5.1), Lemma 5.1, and Lemma 5.2 into (2.4), the displayed estimate is N(σ,T) ≤ K(σ,T0) T^{4σ(1−σ)} log^4 T. The only factor of log T beyond the second-moment bound comes from the prefactor 1/(σ−α) = log T, since α = σ − 1/log T. Hence the first integral contributes (C3(σ,T0)/(2π)) T^{4σ(1−σ)} log^4 T, not log^{5−2σ}. There is no subsequent step converting log^4 into log^{5−2σ}; indeed, for σ > 1/2, 5−2σ < 4, so the advertised exponent is strictly smaller and stronger. The constant K(σ,T0) in (5.3) is defined for the log^4 bound, and the theorem and abstract reuse this K with log^{5−2σ}; this is not a harmless typo because the novelty of the paper is precisely the improved logarithm exponent. A secondary concern is that Lemmas 5.1 and 5.2 are imported without visible hypothesis checks for this particular mollifier, but the exponent mismatch alone invalidates the central claim as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper aims to give the first explicit version of Carlson's zero-density estimate for the Riemann zeta function, claiming N(σ,T) ≤ K(σ,T0) T^{4σ(1−σ)} (log T)^{5−2σ} for T ≥ T0 ≥ 3·10^12 and σ ≥ 0.6, with K defined in (5.3). The proof follows Carlson's method via Littlewood's counting lemma, a mollified zeta function, a second-moment estimate, and explicit bounds from Kadiri–Lumley–Ng. The advertised improvement over Carlson's log^4 T consists of replacing the logarithm exponent 4 by 5−2σ.","tokens_in":8750,"tokens_out":2131,"duration_ms":20087,"significance":"If correct, an explicit Carlson-type estimate with the improved logarithm exponent would be a useful addition to the explicit zero-density literature, complementing recent work by Kadiri–Lumley–Ng, Simonič, and Bellotti. The paper draws on published external lemmas and does not rely on curve fitting or circular reasoning, which is a strength. However, the central advertised exponent is not derived, and the transferred use of two key lemmas is not fully justified; as it stands, the main theorem is unsupported.","major_comments":[{"comment":"The proof concludes with N(σ,T) ≤ K(σ,T0) T^{4σ(1−σ)} log^4 T, not the advertised log^{5−2σ} T. In the final step, the prefactor 1/(σ−α) equals log T, since α = σ − 1/log T, so the first integral contributes at most (C3(σ,T0)/(2π)) T^{4σ(1−σ)} log^4 T. There is no subsequent step that converts log^4 T into log^{5−2σ} T; indeed, for σ ≥ 0.6, 5−2σ < 4, so the claimed exponent is stronger and does not follow. The constant K(σ,T0) in (5.3) is explicitly written for the log^4 T bound, and the theorem and abstract reuse this K with log^{5−2σ} T, which is inconsistent. The main claim of the paper is therefore not established.","section":"§5, Theorem 1.1 and final display"},{"comment":"The sentence 'Carlson obtained as 3' misstates the classical bound (1.2), which has log^4 T. Since the whole novelty of the paper is the improved logarithm exponent, this misstatement compounds the exponent error in Theorem 1.1 and should be corrected or removed.","section":"§4, text near (4.4)"},{"comment":"Lemmas 5.1 and 5.2 are imported from Kadiri–Lumley–Ng without a verification that their hypotheses hold for the particular h_X(s) = ζ(s)M_X(s)(2 − ζ(s)M_X(s)) with X = T^{2σ−1} log T. The statement of Lemma 5.1 contains 'on fixing some parameters' without specifying those choices, and the paper does not show that the required conditions (e.g., the ranges of α, β, η, and the size of X) are satisfied in the present setting. If these bounds do not transfer to this mollifier, the estimates for the argument and vertical-boundary integrals in (2.4) fail. This gap is independent of the exponent mismatch and also affects the final bound.","section":"§5, Lemmas 5.1 and 5.2"}],"minor_comments":[{"comment":"The abstract states a constant 0.78, but Theorem 1.1 gives K(σ,T0) and Table 1 lists several different values; the relationship between 0.78 and K(σ,T0) should be clarified.","section":"Abstract and Theorem 1.1"},{"comment":"There are several typographical issues, including 'seperately' in §4, 'integals' in §5, and inconsistent notation for Simonič's name and reference formatting. These do not affect the mathematics but should be corrected.","section":"Throughout"},{"comment":"The dyadic summation bound leading to the factor 1/(1 − 0.54σ(1−σ)) is plausible, but the origin of the constant 0.54 should be stated explicitly, since it appears to be an input to the summation and is not derived in the text.","section":"§4, equation (4.5) and (4.6)"}],"recommendation":"reject","confidential_remarks":"The central claim of the paper is not proven: the derivation yields log^4 T, not the advertised log^{5−2σ} T, and the improvement over Carlson's classical bound vanishes. This is a load-bearing error that cannot be fixed by local edits without changing the main theorem and its claimed novelty. The additional concern about the unverified transfer of Lemmas 5.1 and 5.2 strengthens the case for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the advertised exponent is not proved. Theorem 1.1 and the abstract claim N(σ,T) ≤ K T^{4σ(1−σ)} (log T)^{5−2σ}, but the proof in §5 ends with (log T)^4. Since the entire novelty over Carlson is the log exponent, the central claim fails as stated.\n\nThe paper is not without merit. The explicit-constant version of Carlson's method appears to be new; I don't know of another published explicit Carlson-type bound. The Littlewood counting and mollifier setup is standard but handled cleanly. The dyadic sum step that replaces the sum of (4.5) by a geometric series with factor 0.54σ(1−σ) is a nice concrete touch, and the table of K(σ,T0) values is useful.\n\nThe soft spots are serious. The exponent mismatch is load-bearing: the proof's K(σ,T0) in (5.3) is defined for the log^4 bound, and reusing it with log^{5−2σ} is not a typo because the whole point is the sharper log power. Relatedly, the remark in §4 that 'Carlson obtained as 3' is wrong; Carlson's bound (1.2) has log^4 T. A second, smaller issue is that Lemmas 5.1 and 5.2 are imported from Kadiri–Lumley–Ng without checking their hypotheses for this specific h_X and X = T^{2σ−1} log T; the phrase 'on fixing some parameters' in Lemma 5.1 is not unpacked. This alone wouldn't sink the paper, but since the main theorem already collapses, it needs attention in any revision.\n\nWho is this for? Explicit analytic number theorists working on zero-density estimates and their applications. They would want to see this fixed.\n\nRecommendation: reject the current version. But I would not bury it. A revision that honestly states the log^4 bound (or actually derives the log^{5−2σ}) and verifies the KLN lemmas would deserve a serious referee.","headline":"The paper advertises a sharper log-exponent than it proves; the proof ends with Carlson's original log^4, so the main claim as stated collapses.","tokens_in":9278,"tokens_out":3849,"would_cite":false,"duration_ms":34599,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N56","11N37","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims an explicit Carlson-type zero-density bound: $N(\\sigma,T) \\le K(\\sigma,T_0)T^{4\\sigma(1-\\sigma)}(\\log T)^{5-2\\sigma}$ for $T\\ge T_0\\ge 3\\cdot 10^{12}$ and $\\sigma\\ge 0.6$.","keywords":["zero density estimates","Riemann zeta function","Carlson's theorem","explicit estimates","mollified zeta function","approximate functional equation","Littlewood's method","nontrivial zeros"],"falsifier":"Check the hypotheses of the two quoted lemmas for $h_X(s)=\\zeta(s)M_X(s)(2-\\zeta(s)M_X(s))$ with $X=T^{2\\sigma-1}\\log T$ and recompute the final display in Section 5: if the chain yields $\\log^4 T$ rather than $(\\log T)^{5-2\\sigma}$, the theorem as stated is not established by the given proof.","tokens_in":8288,"feed_emoji":"🧮","tokens_out":11847,"duration_ms":99320,"temperature":0.7,"pith_summary":"The paper aims to turn Carlson's classical zero-density estimate for the Riemann zeta function into an explicit, usable inequality. It claims that for $T \\ge T_0 \\ge 3\\cdot 10^{12}$ and $\\sigma \\ge 0.6$, the number $N(\\sigma,T)$ of nontrivial zeros with real part greater than $\\sigma$ and imaginary part up to $T$ satisfies $N(\\sigma,T) \\le K(\\sigma,T_0)\\,T^{4\\sigma(1-\\sigma)}(\\log T)^{5-2\\sigma}$, with numerical constants $K(\\sigma,T_0)$ tabulated in the paper. This would be the first explicit version of Carlson's bound, and the logarithmic exponent $5-2\\sigma$ is meant to sharpen Carlson's asymptotic $\\log^4 T$. Zero-density bounds of this kind control how many zeros can lie off the critical line, which matters for results on primes in short intervals and related arithmetic questions. The proof uses Littlewood's zero-counting lemma on a mollified zeta function and bounds the resulting integrals explicitly.","feed_headline":"Carlson's zero-density bound is made explicit for σ≥0.6","feed_subtitle":"The proof supplies usable constants K(σ,T0) and a logarithmic-power improvement over Carlson's bound.","key_machinery":"The central object is the mollified zeta function $h_X(s)=\\zeta(s)M_X(s)(2-\\zeta(s)M_X(s))$, where $M_X(s)=\\sum_{n\\le X}\\mu(n)n^{-s}$ is the Möbius mollifier and $X=T^{2\\sigma-1}\\log T$. Zeros of $\\zeta$ are zeros of $h_X$, so Littlewood's rectangle lemma converts $N(\\sigma,T)$ into integrals of $\\log|h_X|$ and $\\arg h_X$. The main estimate is a second-moment bound $\\int_H^T |f_X(\\alpha+it)|^2\\,dt \\le C_3(\\sigma,T_0)\\,T^{4\\sigma(1-\\sigma)}\\log^3 T$, where $f_X=\\zeta M_X-1$; it is obtained from an explicit approximate functional equation for $\\zeta$ and a mean-value theorem for Dirichlet polynomials, and the remaining argument and vertical-log integrals are handled by two cited lemmas.","core_discovery":"On the paper's own terms, the discovery is that Carlson's asymptotic estimate $N(\\sigma,T)=O(T^{4\\sigma(1-\\sigma)}\\log^4 T)$ can be made fully explicit with a reasonably small constant. Theorem 1.1 asserts that for $\\sigma\\ge 0.6$ and $T\\ge T_0\\ge 3\\cdot 10^{12}$, one has $N(\\sigma,T) \\le K(\\sigma,T_0)\\,T^{4\\sigma(1-\\sigma)}(\\log T)^{5-2\\sigma}$, with $K(\\sigma,T_0)$ defined in (5.3), tabulated for several $T_0$ values, and tending to $\\frac{1}{2\\pi}\\frac{0.68\\,\\sigma(2\\sigma-1)^2}{1-0.54\\,\\sigma(1-\\sigma)}$ as $T_0$ grows. The author presents the result as an improvement in the exponent of the logarithm factor over Carlson's original bound and compares it numerically with the best existing explicit estimates, identifying the range of $\\sigma$ for which the new inequality is sharpest. The intended contribution is a small explicit constant and a logarithmic-power refinement obtained from relatively elementary ingredients: an approximate functional equation, a mean-value estimate for Dirichlet polynomials, and Littlewood's counting argument.","pith_inferences":["A close reading of Section 5 shows that the displayed chain of inequalities ends with a factor $\\log^4 T$, whereas the theorem statement and abstract advertise $(\\log T)^{5-2\\sigma}$; for $\\sigma>1/2$ these two exponents differ, so the advertised logarithmic improvement does not appear in the final displayed calculation. This is an editorial observation about the text, not a verdict on whether a m","If the logarithm exponent is taken to be $4$, the result remains an explicit Carlson-type estimate with explicit constants, and the crossover comparisons with other explicit estimates would need to be recomputed.","A finite test of the method would be to instantiate every displayed inequality in Section 5 for a fixed $\\sigma$ (say $\\sigma=0.6$) and check exactly which power of $\\log T$ the chain yields.","The same mollifier construction with $X=T^{2\\sigma-1}\\log T$ could be tried for other $L$-functions, but portability requires proving the analogue of the two quoted argument and vertical-logarithm bounds for the new $L$-function."],"forward_implications":["For $\\sigma\\in[0.6,2/3]$ and sufficiently large $T$, the new bound becomes sharper than the best near-critical explicit estimate; the paper gives crossover points such as $T\\ge 9.48\\cdot 10^{308}$ for $T_0=3\\cdot 10^{12}$ and $T\\ge 1.43\\cdot 10^{236}$ for $T_0=10^{200}$.","The constants $K(\\sigma,T_0)$ in Table 1 are ready for direct use in applications, and their large-$T_0$ limit has a simple closed form.","Because the proof relies only on the approximate functional equation, mean-value estimates for Dirichlet polynomials, and Littlewood's counting lemma, the same scheme is designed to transfer to Dirichlet $L$-functions and Dedekind zeta functions.","The estimate is strongest for $\\sigma\\le 2/3$, so it fills a region of the $(\\sigma,T)$-plane where earlier explicit bounds are weaker."],"supporting_citations":[{"why":"supplies the verified range $T\\le 3\\cdot 10^{12}$ with no zeros off the critical line, allowing the argument to start at $H_0$.","marker":"[PT20]"},{"why":"is the original Carlson estimate that this paper makes explicit and improves.","marker":"[Car20]"},{"why":"supplies the two lemmas bounding $\\arg h_X$ and $-\\int \\log|h_X|$ that handle three of the four integrals in the zero-counting inequality.","marker":"[KLN18]"},{"why":"provides the weak explicit approximate functional equation used for $\\zeta(s)$ in Lemma 3.2.","marker":"[Sim19]"},{"why":"provides the mean-value theorem for Dirichlet polynomials used in Lemma 3.3.","marker":"[Ram15]"},{"why":"supplies the divisor-sum bound $\\sum_{n\\le t}d(n)^2 \\le \\frac14 t\\log^3 t$ used in the tail estimates.","marker":"[CHT21]"},{"why":"contains Littlewood's classical lemma that expresses the averaged zero count as a contour integral of $\\log h(s)$.","marker":"[Tit86]"}],"fun_headline_variants":["Explicit Carlson bound reveals sharper zero-density estimate","Carlson's zero-density bound made explicit with improved log exponent","New explicit constants for Riemann zeta zero counting","Tight explicit bound on zeta zeros for σ≥0.6","Explicit zero-density theorem beats Carlson's log power"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The estimate collapses if the quoted bounds on the argument and vertical-logarithm integrals do not apply to the specific mollified function $h_X(s)=\\zeta(s)M_X(s)(2-\\zeta(s)M_X(s))$ with $X=T^{2\\sigma-1}\\log T$, and the paper does not verify the stated hypotheses of those quoted lemmas for that function.","fun_headline_variants_meta":{"raw":{"variants":["Explicit Carlson bound reveals sharper zero-density estimate","Carlson's zero-density bound made explicit with improved log exponent","New explicit constants for Riemann zeta zero counting","Tight explicit bound on zeta zeros for σ≥0.6","Explicit zero-density theorem beats Carlson's log power"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00075,"raw_usage":{"total_tokens":3314,"prompt_tokens":893,"completion_tokens":2421,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":2341}},"tokens_in":509,"tokens_out":2421,"duration_ms":16162,"temperature":1.0,"reasoning_tokens":2341,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:52:55.414234+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the hypotheses of the two quoted lemmas for $h_X(s)=\\zeta(s)M_X(s)(2-\\zeta(s)M_X(s))$ with $X=T^{2\\sigma-1}\\log T$ and recompute the final display in Section 5: if the chain yields $\\log^4 T$ rather than $(\\log T)^{5-2\\sigma}$, the theorem as stated is not established by the given proof.","supporting_citations":[],"review_version":1}