REVIEW 3 major objections 2 minor 1 cited by
A new zero-density estimate for $\zeta(s)$ and the error term in the Prime Number Theorem
T0 review · 3 major / 2 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read 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 $\varepsilon=0$.
desk verdict Bold abstract, but the unquantified width in the zero-density bound makes the epsilon=0 conclusion unverifiable as submitted. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [Abstract (first sentence)] 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.
- [Abstract (notation N(σ,T))] 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.
- [Abstract (overall)] 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.
minor comments (2)
- [Abstract (terminology)] 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.
- [Abstract (quantitative clarity)] 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.
Circularity Check
No circularity detected in the abstract: the zero-density estimate and PNT error term are presented as separate results with an external zero-free-region input.
full rationale
The abstract announces a zero-density estimate N(σ,T)=O(1) near the left edge of the Korobov–Vinogradov zero-free region and derives the prime-number-theorem error term from it via the stated optimization ω(x). The zero-free region is introduced as an external known input: 'ν(t)=A_0(log t)^{-2/3}(log log t)^{-1/3} is a decreasing function such that ζ(σ+it)≠0 for σ≥1−ν(t)'. It is not defined in terms of the paper's own output, and no fitted parameter is renamed as a prediction. There is no self-citation chain carrying the argument, and no equation in the visible text reduces to its own input by construction. The unquantified phrase 'sufficiently close to the left edge' is a legitimate correctness concern: if the width of the O(1) band is too small, the claimed ε=0 consequence may not follow. But that is a gap or possible falsehood, not circularity. Because this is an abstract-only review, no hidden derivation can be inspected, but the visible logical structure is a conditional implication from an external zero-free-region hypothesis to a zero-density estimate and then to a PNT error term, not an identity of premise and conclusion.
Assumptions & free parameters
assumptions (1)
- domain assumption Korobov-Vinogradov zero-free region: zeta(sigma+it) != 0 for sigma >= 1 - nu(t), with nu(t)=A_0 (log t)^{-2/3} (log log t)^{-1/3}.
Cite this review
Pith. "Pith review of A new zero-density estimate for $\zeta(s)$ and the error term in the Prime Number Theorem." pith.science (2026). https://pith.science/paper/3LYC2D5Q
@misc{pith2026250802041,
author = {Pith},
title = {Pith review of: A new zero-density estimate for $\zeta(s)$ and the error term in the Prime Number Theorem},
year = {2026},
howpublished = {\url{https://pith.science/paper/3LYC2D5Q}},
note = {Machine review of arXiv:2508.02041}
}
abstract
We will provide a new type of zero-density estimate for $\zeta(s)$ when $\sigma$ is sufficiently close to $1$. In particular, we will show that $N(\sigma,T)$ can be bounded by an absolute constant when $\sigma$ is sufficiently close to the left edge of the Korobov-Vinogradov zero-free region. As a consequence, we provide the optimal error term in the prime number theorem of the form $$ \psi(x)-x \ll x\exp \left\{-(1-\varepsilon) \omega(x)\right\},\qquad \omega(x):=\min _{t \geq 1}\{\nu(t) \log x+\log t\}, $$ where $\nu(t)=A_0(\log t)^{-2/3}(\log\log t)^{-1/3}$ is a decreasing function such that $\zeta(\sigma+it)\neq 0$ for $\sigma\ge 1-\nu(t)$. Precisely, we will show that we can take $\varepsilon=0$.
Forward citations
Cited by 1 Pith paper
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A round of Pintz to celebrate oscillations in sums
An explicit version of Pintz's Landau-type theorem converts bounds on arithmetic sums into zero-free regions, with a worked Mertens/zeta example and a new explicit lower mean bound.
Reviewed August 6, 2026 · model on record in the stance chip above.
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