REVIEW
On Landau's "local method" for obtaining zero-free regions
Number Theory · Submitted 2 Nov 2026 · 27 references
Pith's verdict: A real contribution: the long-folklore Beurling PNT is finally proved with explicit constants, the main theorem is sound, and the soft spots are compressed sections rather than load-bearing flaws.
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Referee report
The paper
The paper claims
Landau's local method is extended to its full generality: a positive-measure Mellin transform with a strip bound gets a zero-free region with explicit constants — giving the Beurling prime number theorem and showing near-1 zeros cluster.
The reading
What carries the argument
Three ingredients carry the argument. The first is Lemma 2.1, a function-theoretic lemma that approximates $f'(z)/f(z)$ inside a disk by a sum over the zeros inside it, with $r$-normalized Blaschke factors $B_r(z;\rho)$ supplying the correction terms; it converts the local bound $|F| \le e^{\varphi(t)}$ into quantitative control of $F'/F$ at the sample points. The second is Landau's extremal problem, the infimum $V$ of $(a_1 + \cdots + a_N)/(\sqrt{a_1} - \sqrt{a_0})^2$ over nonnegative cosine polynomials $P(x) = a_0 + \sum_{n=1}^N a_n \cos(nx)$ with $0 < a_0 < a_1$ and $a_n \ge 0$; the known bounds $34.468305 < V < 34.4889920009$ fix the explicit constants in the zero-free region and PNT statements. The third, the engine of the proof, is the positivity identity (3.7): the real part of the linear combination $-a_0 F'(\sigma_0)/F(\sigma_0) - \sum_{n=1}^N a_n F'(s_n)/F(s_n)$ equals $\int_1^{\infty} (\log x)\, P(\gamma_0 \log x)\, x^{-\sigma_0}\, dm(x) \ge 0$, nonnegative because the measure is positive and $P$ is nonnegative on the real line. Bounding this nonnegative quantity from above with the zero-sum lemma, the pole bounds, and the strip bound $\varphi$ forces any near-1 zero $\rho_0$ to satisfy $1 - \beta_0 \ge (\sqrt{a_1} - \sqrt{a_0})^2/(2(\alpha a_0 + a_1 + \cdots + a_N)) \cdot 1/(\varphi(\gamma_0)\psi(\gamma_0))$, which after optimizing over the polynomial and the parameter $\kappa$ yields the constant $1/(2(1+\alpha)V)$. The clustering section runs the same structure with Mertens' polynomial, relying on the identity $3 + 4\cos x + \cos 2x = 2(1+\cos x)^2 \ge 0$.
Extended reading notes
Core claim
The load-bearing assertion is Theorem 3.1. Let $F(s) = \exp(\int_1^{\infty} x^{-s}\,dm(x))$ for a positive measure $dm$, continued analytically to the strip $\sigma \ge 1 - 1/\psi(|t|)$ except for a simple pole at $s = 1$, with the pole-side estimates $F(\sigma) \le A/(\sigma-1)$ and $-F'(\sigma)/F(\sigma) \le 1/(\sigma-1) + B$ for $1 < \sigma \le 2$, and with the uniform bound $|F(s)| \le e^{\varphi(t)}$ throughout the strip; here $\varphi, \psi$ are positive, non-decreasing, slowly varying, and $\psi(t) \ll e^{(\alpha+o(1))\varphi(t)}$. Then for every $c > 1/(2(1+\alpha)V)$ the function $F$ has no zeros in the region $\sigma \ge 1 - c/(\psi(t)\varphi(t))$, $t \ge t_0$, where $V$ is Landau's extremal constant from the theory of nonnegative cosine polynomials. The theorem works because the positivity of the measure, not a functional equation, drives the argument, so it applies exactly where the local method is needed: Beurling zeta functions, which possess no analytic continuation machinery beyond a strip. Applying it with $\psi(t) = 2/(1-\theta)$ and $\varphi(t) = (1/2+\varepsilon)\log t + O(1)$ to systems with $N(x) = Ax + O(x^{\theta})$ gives Corollary 3.3: no zeros for $\sigma > 1 - (1/(2V) - \varepsilon)(1-\theta)/\log|t|$, hence the prime number theorem with error $O(x\exp(-c'\sqrt{\log x}))$ for $c' < \sqrt{2(1-\theta)/V}$. The same hypotheses are then shown to control the local number of zeros with explicit constants, to extend, for almost any $b$, Littlewood's short-interval zero-counting under the Riemann hypothesis to settings without a functional equation, and to force any zero close to the 1-line to come with a cluster of nearby zeros; feeding the clustering information back into the main estimate shows the constant $1/(2W_\alpha)$ is not sharp.
Load-bearing premise
The load-bearing premise is the strip input itself: that $F$ continues analytically to $\sigma \ge 1 - 1/\psi(|t|)$ and obeys $|F(s)| \le e^{\varphi(t)}$ there, with $\psi$ growing no faster than $e^{(\alpha+o(1))\varphi(t)}$; the paper assumes this data rather than produces it, and for the Beurling corollary it is imported from cited estimates in [21] and [24] — if the strip is narrower or the bound weaker, neither the zero-free region nor the PNT error term follows.
Editorial extensions
If this is right
- For Beurling number systems with well-behaved integers, $N(x) = Ax + O(x^{\theta})$ with $\theta \in [0,1)$, the prime number theorem holds with error $O(x\exp(-c'\sqrt{\log x}))$ for every $c' < \sqrt{2(1-\theta)/V} \approx 0.24\sqrt{1-\theta}$; the paper supplies the proof of a zero-free region and PNT error term that earlier sources had mentioned without proof.
- Under the same hypotheses, the number of zeros in a disk of radius $R$ about $1 + it$ with $R\,\psi(t) \le a < 1$ is at most $C(a)(1+\alpha+o(1))\,R\,\varphi(t)\psi(t) + 3$ with an explicit $C(a)$ — up to constants, exactly the zero density permitted by the zero-free region.
- For the Riemann zeta function under the Riemann hypothesis, vertical intervals of length $R \gg (\log\log\log\tau)/\log\log\tau$ contain $O(R\log\tau)$ zeros, falling short of Littlewood's range $R \gg 1/\log\log\tau$ only by a triple-log factor and obtained without the Riemann–von Mangoldt formula.
- A zero $\rho_0 = \beta_0 + i\gamma_0$ with $1 - \beta_0 = \nu/(\varphi(\gamma_0)\psi(\gamma_0))$ and $\nu \ge 1/(4(1+\alpha)V)$ is never isolated: for radii $R$ between $c_1\nu^2/(\varphi\psi)$ and $c_2/\psi$, the pair of disks around $1 + i\gamma_0$ and $1 + 2i\gamma_0$ contains at least $1 + C R\nu^2\varphi(\gamma_0)\psi(\gamma_0)$ zeros.
- Combining clustering with the main estimate yields a positive $\delta(\alpha)$ such that every large-height zero satisfies $1 - \beta \ge (1/(2W_\alpha) + \delta)/(\varphi(\gamma)\psi(\gamma))$; the constant produced by the plain form of the method is provably not sharp, and the paper leaves the optimal constant (conjecturally $c(\varphi,\psi) = 1$, with a refined guess $1/(1+\alpha)$) as an open
Reading between the lines
- If the paper's conjecture that the optimal constant is $c(\varphi,\psi) = 1$ (refined guess $1/(1+\alpha)$) is right, the Beurling prime-number constant would improve from $c' < \sqrt{2(1-\theta)/V} \approx 0.24\sqrt{1-\theta}$ to $c' < 2\sqrt{1-\theta}$, a factor near eight; the Section 6 feedback mechanism is a concrete route toward that gap.
- Theorem 3.5 already sketches a character-twisted version of the positivity argument; a natural test is to instantiate it with Dirichlet characters, whose twisted transforms satisfy strip bounds by classical results, and compare the resulting zero-free constant with the best known de la Vallée Poussin constant for $L$-functions.
- The paper itself notes that the $n(1 + 2i\gamma_0, R)$ term and the quadratic $\nu^2$ clustering dependence are by-products of using a degree-2 cosine polynomial; a degree-$N$ polynomial refinement would trade in cluster points $n(1 + ki\gamma_0, R)$ for a sharper $\nu$-dependence, and the cited linear-in-$\nu$ result for Beurling systems suggests the quadratic law is not the final word.
- Because the Turán-type section never uses an asymptotic zero-counting formula, the same short-interval bound should hold for any zeta-like function satisfying only the local strip hypotheses — for instance Selberg-type or Epstein zeta functions, where no Riemann–von Mangoldt analogue exists.
Original abstract
We extend Landau's local analytic method for the deduction of zero-free region to a very general setting, and as a particular consequence, obtain a zero-free region and corresponding Prime Number Theorem for Beurling number systems with well-behaved integers. We also discuss the sharpness of the method and prove estimates for the local number of zeros, as well as a clustering result, which roughly put says that zeros close to the 1-line cannot be isolated but must occur in clusters.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We extend Landau's local analytic method for the deduction of zero-free region to a very general setting, and as a particular consequence, obtain a zero-free region and corresponding Prime Number Theorem for Beurling number systems with well-behaved integers. We also discuss the sharpness of the method and prove estimates for the local number of zeros, as well as a clustering result, which roughly put says that zeros close to the 1-line cannot be isolated but must occur in clusters.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We extend Landau's local analytic method for the deduction of zero-free region to a very general setting, and as a particular consequence, obtain a zero-free region and corresponding Prime Number Theorem for Beurling number systems with well-behaved integers. | contribution 0.99 | 0.95 | 0.81 | asserted 1.00 |
| s1 | We also discuss the sharpness of the method and prove estimates for the local number of zeros, as well as a clustering result, which roughly put says that zeros close to the 1-line cannot be isolated but must occur in clusters. | contribution 0.52 | 0.95 | 0.13 | asserted 0.99 |
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References
Discussion
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