{"id":"871b72b9-1e61-40ff-a7f0-8781b6bb2c8f","arxiv_id":"2508.02041","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that the number of zeta zeros with real part very close to 1 is bounded by an absolute constant, leading to an optimal error term in the prime number theorem with epsilon=0.","lead":"This paper claims a new zero-density estimate for the Riemann zeta function that bounds the number of zeros near the 1-line, and uses it to improve the error term in the prime number theorem to the best possible form allowed by the Korobov-Vinogradov zero-free region. A generalist might read it because the prime number theorem is a central result, and any improvement in its error term has consequences across number theory.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The ε=0 PNT conclusion hinges on an unquantified 'sufficiently close' width in the zero-density estimate; if that width is o(ν(t)), the claimed error does not follow from the stated bound.","rationale":"The reader correctly identified the uniformity of the zero-density band as the weakest assumption. My stress-test sharpens that concern: it is not merely that the width must not shrink to zero with T; to deliver ε=0, the width must be at least a specific positive multiple of ν(t) at the optimizing height. This is because the minimization in ω(x) balances ν(t)log x against log t, and a bounded zero count at the band edge can only be discounted against the desired error if the band width is large enough to absorb the log t term. The abstract does not specify this width, so the central claim is unverifiable from the available material. I do not reject the paper; the mathematical claim may well be correct, and the known Korobov-Vinogradov zero-free region makes the existence of such a band plausible. However, without the proof's explicit constants and range of σ, the ε=0 consequence cannot be checked. The absence of full text also means no independent verification or machine-checked support is available. Thus the reader's UNVERDICTED verdict remains unchanged; my concern shows one concrete condition that would need to be satisfied for the central claim to hold, but it does not by itself establish a flaw.","tokens_in":831,"tokens_out":16793,"duration_ms":226353,"concrete_test":"In the full manuscript, locate the exact statement of the zero-density estimate and extract the explicit range of σ for which N(σ,T)≤C. In particular, determine whether the theorem gives N(σ,T)≤C for σ ≥ 1-(1+δ)ν(T) with an absolute constant δ ≥ 2/3, or only for a fixed absolute distance δ0, or only for a T-dependent width. Then take t0 minimizing ν(t)log x+log t for large x and substitute the established range into the standard Perron-contour estimate for ψ(x)-x; verify that the total contribution from zeros in the band is at most x exp(-ω(x)). If the stated range gives δ < 2/3 at t0 or a width o(ν(t)), recompute the resulting ψ-error and check whether it is still x exp(-ω(x)). This check directly decides whether the claimed ε=0 follows from the proven zero-density statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract leaves unquantified what 'sufficiently close to the left edge' of the Korobov-Vinogradov zero-free region means for the new zero-density estimate. The PNT consequence with ε=0 only follows if the band in which N(σ,T)=O(1) has a width η(t) at least about (2/3)ν(t) uniformly in T. At the minimizer t0 of ω(x)=min_t{ν(t)log x+log t}, the stationary condition gives log t0 ≈ (2/3)ν(t0)log x, so ω(x) ≈ (5/3)ν(t0)log x. A bounded number of zeros at the left edge of the band (σ=1-ν(t0)-η(t0)) contributes at most O(1)·x exp(-(ν(t0)+η(t0))log x). To keep this below x exp(-ω(x)) = x exp(-(5/3)ν(t0)log x), one needs η(t0) ≥ (2/3)ν(t0). If the proof only gives a shrinking width, for instance η(t)=o(ν(t)) or η(t)∼1/log t, then the exceptional zero contribution exceeds the claimed error term, and the ε=0 conclusion does not follow. The abstract's vagueness on this uniformity is therefore a load-bearing gap, not merely a cosmetic one. Because no proof is available for inspection, this condition cannot currently be verified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a new zero-density estimate for the Riemann zeta function: when σ is sufficiently close to the left edge of the Korobov–Vinogradov zero-free region, the count N(σ,T) of zeros with real part at least σ and imaginary part at most T is bounded by an absolute constant. From this it derives the Prime Number Theorem error term ψ(x)−x ≪ x exp{−(1−ε)ω(x)} with the displayed function ω(x) defined in the abstract, and claims that ε=0 is attainable. The abstract states these results but provides no proof, no equations beyond the statement, and no quantitative description of the region 'sufficiently close' to the zero-free boundary.","tokens_in":1118,"tokens_out":8469,"duration_ms":84655,"significance":"If the claims are correct, this is a major advance: it would give an error term in the Prime Number Theorem of the form x exp{−ω(x)} with no loss of the (1−ε) factor, which is the best currently obtainable from the Korobov–Vinogradov zero-free region. The underlying zero-density statement, N(σ,T)=O(1) in a band bordering the zero-free region, would be far stronger than known density estimates, which typically give growth like T^{c(1−σ)} or exp{o(log T)}. The paper's statements are precise and falsifiable, and the connection between the zero-density estimate and the PNT error term is explicit. However, the extraordinary strength of the zero-density claim places the burden of proof very high, and the abstract alone provides no material to verify it.","major_comments":[{"comment":"The phrase 'sufficiently close to the left edge of the Korobov–Vinogradov zero-free region' is unquantified. This is load-bearing because the claimed ε=0 PNT error term requires a specific lower bound on the width of the band where N(σ,T)=O(1). Writing the band as σ=1−ν(t)−η(t), the contribution of a bounded number of zeros at its left edge is at most O(1)·x exp{−(ν(t)+η(t))log x}. At the stationary point t0 of ω(x), where log t0 ≈ (2/3)ν(t0)log x, one has ω(x) ≈ (5/3)ν(t0)log x; to keep the exceptional-zero contribution below x exp{−ω(x)} one needs η(t0) ≥ (2/3)ν(t0). The abstract does not state that the allowed closeness is proportional to ν(t), nor that it is uniform in T. If the proof only gives a width o(ν(t)), the claimed error term does not follow from the stated bound. The authors must specify the quantitative width of the band and its uniformity in T.","section":"Abstract (first sentence)"},{"comment":"The notation N(σ,T) conventionally denotes a count with a fixed σ. If σ is fixed, then 'sufficiently close to the left edge' can only mean a fixed horizontal strip of some absolute width η>0, and the claim asserts N(1−η,T)=O(1) for an absolute η. That is an extraordinarily strong and unproved assertion. If instead σ is allowed to depend on T (or on the height t of the zero), then the abstraction of N(σ,T) as a function of a fixed σ is misleading, and the precise T-dependence and uniformity of the absolute constant must be stated. The PNT consequence depends on this uniformity, so the paper must clarify which interpretation is intended and give the exact condition.","section":"Abstract (notation N(σ,T))"},{"comment":"No derivation, proof sketch, or quantitative error estimates are supplied. For a claim of this strength — a uniform constant bound in a region bordering the zero-free zone — known density estimates in comparable regions produce at best exp{o(log T)} or power-of-T bounds, so the asserted O(1) is a qualitative departure from the existing literature. The abstract does not provide any intermediate statements that could be checked, and the reader cannot assess whether the leap from the zero-free region to the PNT error term is justified. The full manuscript must contain a complete proof, and at minimum the abstract should state the precise zero-density result that is being claimed.","section":"Abstract (overall)"}],"minor_comments":[{"comment":"The phrase 'optimal error term' is used without qualification. The bound x exp{−ω(x)} is not known to be optimal in the sense of a matching lower bound for ψ(x)−x; it is better described as the best error term currently derivable from the Korobov–Vinogradov zero-free region.","section":"Abstract (terminology)"},{"comment":"Even in an abstract, the phrase 'sufficiently close' could be made more precise by indicating whether the allowed band has constant width, width proportional to ν(t), or some other scaling; this would materially help the reader judge the implication for the PNT error term.","section":"Abstract (quantitative clarity)"}],"recommendation":"major_revision","confidential_remarks":"The submission currently consists of an abstract only. If that is the entire manuscript, it is not sufficient for refereeing, and the paper should be returned for full submission. If the full paper exists, the refereeing process must focus on the quantitative zero-density statement and its uniformity, because the ε=0 PNT conclusion depends on a specific lower bound for the width of the band. The claim N(σ,T)=O(1) near the zero-free boundary is so strong that I would recommend a very careful check of the proof once it is available; the abstract alone does not provide enough to verify it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is an interesting claim, but it is an abstract, not a paper. If the full proof exists and actually proves that N(sigma,T) is bounded by an absolute constant on a band of width comparable to nu(t) to the left of the Korobov–Vinogradov region, that would be a real advance and the epsilon=0 PNT error term would follow. But as submitted, none of that is checkable.\n\nWhat the abstract does well: it states a sharp, concrete target—epsilon=0 in the omega(x) error term—and correctly connects it to a zero-density estimate. The definition of omega(x) via the minimum over t of nu(t) log x + log t is the standard KV optimization, and the claim is coherent.\n\nThe soft spots are severe. The phrase 'sufficiently close to the left edge' is doing all the work. The stress-test note is right: to get epsilon=0, you need the width of the O(1)-zero band, call it eta(t), to satisfy eta(t) >= (2/3) nu(t) at the minimizer t0. If the proof only gives eta(t) = o(nu(t)) or something like 1/log t, the exceptional zeros in that band contribute more than x exp(-omega(x)), and the claimed error term does not follow. The abstract does not quantify eta(t) at all, so we cannot tell whether the main theorem is even strong enough to deliver the advertised consequence.\n\nAlso, the abstract cites no prior work. I have no way to tell whether the epsilon=0 statement is new or a repackaging of known consequences of KV. The reader's take is fair: novelty is plausible but unverified. There is no derivation, no constants, no uniformity hypothesis beyond 'absolute constant.' For a result this strong, that is not enough.\n\nMy recommendation: if the full paper exists, it deserves a serious referee—the claim is important enough to spend referee time on. But this submission, as it stands, is an abstract with no proof. I would not send it to peer review; I would desk reject and invite the author to submit the full manuscript with the width condition made explicit. The burden is on the author to show eta(t0) >= (2/3)nu(t0) uniformly in T.\n\nNot something I can cite yet.","headline":"Bold abstract, but the unquantified width in the zero-density bound makes the epsilon=0 conclusion unverifiable as submitted.","tokens_in":1642,"tokens_out":2866,"would_cite":false,"duration_ms":25463,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11M06","11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that $N(\\sigma,T)$ is $O(1)$ in a fixed strip left of the Korobov-Vinogradov zero-free region, and derives from it the optimal prime number theorem error term with $\\varepsilon=0$.","keywords":["Riemann zeta function","zero-density estimate","prime number theorem","error term","Korobov-Vinogradov zero-free region","psi(x)","zeta zeros"],"falsifier":"Find a sequence $T_n\\to\\infty$ and a fixed $\\delta>0$ such that the number of zeros of $\\zeta(s)$ in the rectangle $[1-\\nu(T_n)-\\delta, 1-\\nu(T_n)]\\times[0,T_n]$ grows without bound; this would contradict the asserted uniform boundedness of $N(\\sigma,T)$ and invalidate the $\\varepsilon=0$ conclusion.","tokens_in":629,"feed_emoji":"","tokens_out":6390,"duration_ms":60664,"temperature":0.7,"pith_summary":"This paper sets out to prove a zero-density estimate of a new type for the Riemann zeta function: the number $N(\\sigma,T)$ of zeros with real part at least $\\sigma$ and imaginary part at most $T$ is bounded by an absolute constant, provided $\\sigma$ lies in a fixed strip immediately to the left of the Korobov-Vinogradov zero-free region. The payoff is the error term in the prime number theorem, $\\psi(x)-x \\ll x\\exp\\{-(1-\\varepsilon)\\omega(x)\\}$, and the paper claims the exponent can be taken with $\\varepsilon=0$. A sympathetic reader would care because this would remove the customary epsilon loss and attain the strongest known quantitative form of the prime number theorem.","feed_headline":"Zero count near the 1-line is uniformly bounded; PNT error is optimal","feed_subtitle":"A new zero-density estimate removes the epsilon loss in the error term for psi(x)-x, yielding the strongest known form.","key_machinery":"The named objects are the zero-free-region function $\\nu(t)$ and the count $N(\\sigma,T)$. The argument works by controlling the number of zeros inside the narrow strip between the zero-free region and a fixed line to its left; the new density estimate turns the contribution of that strip into an absolute constant, which is exactly what removes the epsilon in the error term.","core_discovery":"The central claim is that $N(\\sigma,T)$ is $O(1)$ uniformly in $T$ when $\\sigma$ is sufficiently close to $1-\\nu(t)$, the left edge of the Korobov-Vinogradov zero-free region, where $\\nu(t)=A_0(\\log t)^{-2/3}(\\log\\log t)^{-1/3}$. Exploiting this uniform boundedness in the minimization that defines $\\omega(x)$, the paper obtains $\\psi(x)-x \\ll x\\exp\\{-\\omega(x)\\}$ with no $\\varepsilon$ slack, i.e. $\\varepsilon=0$ in the stated bound.","pith_inferences":["Editorial inference: the same mechanism would likely transfer to other $L$-functions with a Korobov-Vinogradov zero-free region, sharpening the error term for primes in arithmetic progressions.","Editorial inference: the absolute constant bound suggests zeros just left of the zero-free region are extremely sparse; one could test numerically whether the count is in fact $0$ or $1$ for all large $T$.","Editorial inference: the method may also sharpen the error term for $\\psi(x+h)-\\psi(x)-h$ in short intervals, where a similar epsilon loss currently appears."],"forward_implications":["The optimal error term $\\psi(x)-x \\ll x\\exp\\{-\\omega(x)\\}$ holds with $\\varepsilon=0$.","For any fixed $\\sigma$ to the left of $1-\\nu(t)$ within the stated closeness, $N(\\sigma,T)=O(1)$ uniformly in $T$.","The bound on $N(\\sigma,T)$ is absolute, so it does not degrade as $T$ increases.","Combining the density estimate with known explicit zero-free-region constants gives a fully numerical form of the prime number theorem error term."],"supporting_citations":[],"fun_headline_variants":["Zero-density bound makes PNT error term epsilon-free","Uniform zero count near 1-line yields optimal PNT error","Optimal error term for psi(x)-x via zero-density estimate","Epsilon removed from PNT error by new zero-density estimate","Korobov-Vinogradov edge zero count constant, PNT error optimal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument relies on the Korobov-Vinogradov zero-free region holding with an absolute constant $A_0$, and on the strip where $N(\\sigma,T)$ is bounded having width independent of $T$; if that width shrinks as $T$ grows, the claimed $\\varepsilon=0$ error term would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Zero-density bound makes PNT error term epsilon-free","Uniform zero count near 1-line yields optimal PNT error","Optimal error term for psi(x)-x via zero-density estimate","Epsilon removed from PNT error by new zero-density estimate","Korobov-Vinogradov edge zero count constant, PNT error optimal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001124,"raw_usage":{"total_tokens":4634,"prompt_tokens":864,"completion_tokens":3770,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":3680}},"tokens_in":480,"tokens_out":3770,"duration_ms":28823,"temperature":1.0,"reasoning_tokens":3680,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:11:34.826781+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a sequence $T_n\\to\\infty$ and a fixed $\\delta>0$ such that the number of zeros of $\\zeta(s)$ in the rectangle $[1-\\nu(T_n)-\\delta, 1-\\nu(T_n)]\\times[0,T_n]$ grows without bound; this would contradict the asserted uniform boundedness of $N(\\sigma,T)$ and invalidate the $\\varepsilon=0$ conclusion.","supporting_citations":[],"review_version":1}