REVIEW
Diophantine approximation by primes and Landau--Siegel zeros
Number Theory · Submitted 24 Sep 2026 · 26 references
Pith's verdict: Genuinely new conditional bound at 1/3+ beyond in the inhomogeneous prime problem, soundly built, but Theorem 1.2 hangs on an unverified normalization interface to Merikoski's Theorem 16 that no part of this paper independently checks.
T0 · AI review
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Referee report
The paper
The paper claims
Assuming infinitely many Landau–Siegel zeros, the paper pushes the inhomogeneous prime-approximation exponent from the unconditional 9/28 to 1/3, and under zeros of strength 7 to about 0.3836, concluding that exceptional zeros, if they…
The reading
What carries the argument
The engine is the exact identity (2.1), $\Lambda(n)=\lambda'(n)-\sum_{k\ell=n,\,k>1}\lambda(k)\Lambda(\ell)$, with $\lambda=1*\chi$ and $\lambda'=\chi*\log$ for the exceptional character $\chi$: it converts the prime von Mangoldt function into non-negative divisor-like functions, so the contribution of the convolution term after pre-sieving is shown negligible. The finite-type hypothesis on $\alpha$ enters only through the Diophantine synchronization Lemma 5.1, which forces analytic convergent denominators into multiplicative windows $[D_j^{T_j},\,c^{-1}D_j^{t_\alpha T_j}]$ so that the scale $x=D^V$ makes the error $E=V^{16}\xi^{-1}+\exp(-c\sqrt{V\log\xi})$ vanish; the lower bound past one-third runs on the exceptional-character sieve, whose Type I and reference-sequence hypotheses are verified in Propositions 4.1 and 4.2.
Extended reading notes
Core claim
The central claim is Theorem 1.2: assume infinitely many Siegel zeros of strength 7, meaning real zeros of $L(s,\chi_j)$ at $1-\beta_j = o((\log D_j)^{-7})$ with escalating conductors $D_j$. Then for every irrational $\alpha$ of finite type, every real $\beta$, and every $0<\eta<\eta_0=(\sqrt{e}-1)/(3(1+2\sqrt{e}))=0.0503183978\ldots$, there are infinitely many primes $p$ with $\|\alpha p+\beta\|\le p^{-1/3-\eta}$. The companion Theorem 1.1, under merely infinitely many Siegel zeros, gives the same with exponent $1/3+\varepsilon$. The argument proceeds through an asymptotic formula at the one-third scale (Theorem 3.1) and a lower bound past it (Theorem 4.1), each derived from the exceptional-character identity and sieve, with the improvement quantified by the constant $c(\gamma)=1-2\log((1+3\gamma)/(1-6\gamma))$.
Load-bearing premise
The results are conditional on the unproved existence of infinitely many Siegel zeros, strength 7 for the beyond-one-third piece; additionally, the finite-type growth of $\alpha$'s convergents must place a denominator inside the synchronization window so the error $E$ decays, and the lower-bound theorem depends on the correctness of the cited exceptional-character sieve normalization, whose printed factor the paper flags as a typo.
Editorial extensions
If this is right
- Merely infinitely many Siegel zeros put the inhomogeneous problem at the same one-third exponent attained in the homogeneous case.
- Strength-7 zeros push the exponent to $1/3+\eta$ for every $\eta<\eta_0$, about $0.3836$; the abstract's $p^{-1/3-1/20}$ is the special case $\eta=1/20$.
- Because $c(\gamma)$ stays positive exactly for $\gamma<\eta_0$, the constant $\eta_0$ is the natural ceiling of this sieve method rather than an artifact of one step.
- Every irrational of finite type qualifies, and algebraic irrationals have type 1, so the theorems cover all quadratic-surd-type $\alpha$.
Reading between the lines
- The gap between the abstract's exponent $1/3+1/20$ and the theorem's full $1/3+\eta_0$ suggests the strength-7 construction is near-optimal within this sieve, and a different normalization might move $\eta_0$ itself.
- A testable boundary: for $\alpha$ with large finite type $t_\alpha$, the multiplicative window in Lemma 5.1 may be empty for all strengths $k$, restricting the theorems in practice to type close to 1.
- If the flagged normalization in the cited sieve is wrong, Theorem 4.1's constant $c(\gamma)$ and the whole beyond-one-third claim would need re-derivation; the paper itself points at this risk.
Original abstract
Let $\alpha>0$ be an irrational number of finite type and let $\beta\in\mathbb{R}$. Assuming the infinitude of Siegel zeros (respectively, sufficiently strong Siegel zeros), we show that for every sufficiently small $\varepsilon>0$, there exist infinitely many primes $p$ such that \begin{gather*} \|\alpha p+\beta\|\leq p^{-1/3+\varepsilon} \qquad\text{(respectively, }\|\alpha p+\beta\|\leq p^{-1/3-1/20}\text{)}. \end{gather*}
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
Let $\alpha>0$ be an irrational number of finite type and let $\beta\in\mathbb{R}$. Assuming the infinitude of Siegel zeros (respectively, sufficiently strong Siegel zeros), we show that for every sufficiently small $\varepsilon>0$, there exist infinitely many primes $p$ such that \begin{gather*} \|\alpha p+\beta\|\leq p^{-1/3+\varepsilon} \qquad\text{(respectively, }\|\alpha p+\beta\|\leq p^{-1/3-1/20}\text{)}. \end{gather*}
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\alpha>0$ be an irrational number of finite type and let $\beta\in\mathbb{R}$. | background 1.00 | 0.06 | 0.02 | unclear 0.67 |
| s1 | Assuming the infinitude of Siegel zeros (respectively, sufficiently strong Siegel zeros), we show that for every sufficiently small $\varepsilon>0$, there exist infinitely many primes $p$ such that \begin{gather*} \|\alpha p+\beta\|\leq p^{-1/3+\varepsilon} \qquad\text{(respectively, }\|\alpha p+\beta\|\leq p^{-1/3-1/20}\text{)}. \end{gather*} | result 0.96 | 0.93 | 0.95 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
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- P abstract
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- T body sentences
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References
Discussion
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