REVIEW
An explicit elementary proof of no Landau--Siegel zeros
Number Theory · Submitted 21 Oct 2026 · 13 references
Pith's verdict: A careful, explicit elementary proof for Landau–Siegel zero gap with constant 1/100, building on recent Frobenius method works; soft spots are the numeric inequality verification, but the logic holds together.
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Referee report
The paper
The paper claims
An elementary proof establishes that Landau–Siegel zeros cannot exist, with an explicit gap constant of 1/100.
The reading
What carries the argument
The proof rests on three interacting engines: (1) an arithmetic bound on $S_-(x,\chi)$ via the injectivity of an evaluation map $\iota$ on integer-coefficient polynomials, leveraging a triangular basis change (Lemma 3.2, Lemma 3.3) and prime-power divisibility coming from the congruence $z^p \equiv w \pmod{p}$ at the grid points; (2) a paired-tail hyperbola estimate (Proposition 4.1) that controls $H_\beta(t)$ with an error $O((q+\log t)t^{1/2-\beta})$ free of a $1/(1-\beta)$ factor, achieved by combining two tail series before estimating; and (3) a logarithmic weight comparison (Lemma 5.1) that eliminates any dependence on a lower bound for $L(1,\chi)$ by using the positivity of $a(n)$ and an injection $(n,p)\mapsto np$.
Extended reading notes
Core claim
Theorem 1.1: For every primitive quadratic Dirichlet character $\chi$ of conductor $q \ge 3$, any real zero $\beta \in (0,1)$ of $L(s,\chi)$ satisfies $(1-\beta)\log q > 1/100$. The proof proceeds in three main parts. The arithmetic input (Proposition 3.1) gives an explicit upper bound for $S_-(x,\chi) = \sum_{p\le x,\, \chi(p)=-1} (\log p)/p$, obtained via an injective evaluation map on integer-coefficient polynomials and a triangular basis that supplies prime-power divisibility. The real-variable part (Proposition 4.1) estimates the sum $H_\beta(t) = \sum_{n\le t} a(n)/n^\beta$ at a real zero, with an error term that avoids a factor $1/(1-\beta)$ by pairing the tails of two series. Finally, Lemma 5.1 converts these estimates into a logarithmically weighted prime-sum comparison; assuming $(1-\beta)\log q \le 1/100$ leads to an explicit numerical contradiction. The same argument with adjusted exponents (Appendix A) yields the stronger constant $1/64$. The entire proof uses only elementary methods and a single set of parameters for all conductors.
Load-bearing premise
The constant $1/100$ in Theorem 1.1 depends critically on the existence of an odd prime $p_0$ with $\chi(p_0)=-1$ satisfying $p_0 \ge 10000$ and $p_0 \ge 4q$ when such a prime is the largest contributing one; the explicit numerical bounds derived under those conditions drive the final contradiction, and the small-case covering arguments are separate but consistent.
Editorial extensions
If this is right
- Landau–Siegel zeros are definitively ruled out for all primitive quadratic Dirichlet characters with an explicit uniform gap of $1/100$ in the zero-free region.
- The proof is fully elementary, requiring no analytic continuation, complex analysis, or functional equation, and uses only real summation, polynomial arithmetic, and standard inequalities.
- The constant $1/100$ can be improved to $1/64$ by adjusting the exponent choices (Appendix A), and further optimization is possible without changing the elementary framework.
- All numerical constants in the proof are explicit, making the argument verifiable by direct calculation and independent of any unproven hypotheses.
Reading between the lines
- The paired-tail technique in Proposition 4.1, which cancels the $1/(1-\beta)$ factor, may be applicable to other problems in analytic number theory where a zero of an $L$-function leads to large errors in truncated sums.
- The elementary nature of the proof suggests that similar zero-free region estimates could be obtained for other families of $L$-functions, such as those attached to higher-degree characters or automorphic forms, using analogous polynomial congruences and coefficient counting.
- The parameter choices $(q^6, q^{70})$ are not optimized; a systematic search over exponents could yield a larger constant than $1/64$ while retaining the elementary framework.
Original abstract
We give an elementary proof of the absence of Landau--Siegel zeros with the explicit constant $c=1/100$. For every primitive quadratic Dirichlet character $\chi$ of conductor $q\ge3$, each real zero $\beta\in(0,1)$ of $L(s,\chi)$ satisfies $(1-\beta)\log q>c$. The proof combines integer coefficient counting, polynomial congruences, a paired-tail hyperbola estimate, and a logarithmically weighted comparison of positive sums. All required constants are explicit, and one choice of parameters covers every conductor. The argument uses elementary arithmetic and real summation and integration, without separate zero computations for small conductors.
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 give an elementary proof of the absence of Landau--Siegel zeros with the explicit constant $c=1/100$. For every primitive quadratic Dirichlet character $\chi$ of conductor $q\ge3$, each real zero $\beta\in(0,1)$ of $L(s,\chi)$ satisfies $(1-\beta)\log q>c$. The proof combines integer coefficient counting, polynomial congruences, a paired-tail hyperbola estimate, and a logarithmically weighted comparison of positive sums. All required constants are explicit, and one choice of parameters covers every conductor. The argument uses elementary arithmetic and real summation and integration, without separate zero computations for small conductors.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We give an elementary proof of the absence of Landau--Siegel zeros with the explicit constant $c=1/100$. | contribution 0.68 | 0.96 | 0.96 | asserted 1.00 |
| s1 | For every primitive quadratic Dirichlet character $\chi$ of conductor $q\ge3$, each real zero $\beta\in(0,1)$ of $L(s,\chi)$ satisfies $(1-\beta)\log q>c$. | result 1.00 | 0.85 | 0.61 | asserted 1.00 |
| s2 | The proof combines integer coefficient counting, polynomial congruences, a paired-tail hyperbola estimate, and a logarithmically weighted comparison of positive sums. | method 1.00 | 0.92 | 0.03 | asserted 1.00 |
| s3 | All required constants are explicit, and one choice of parameters covers every conductor. | contribution 0.64 | 0.92 | 0.05 | asserted 1.00 |
| s4 | The argument uses elementary arithmetic and real summation and integration, without separate zero computations for small conductors. | method 0.97 | 0.92 | 0.03 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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