Explicit bounds on exceptional zeroes of Dirichlet L-function II
Pith reviewed 2026-05-24 19:31 UTC · model grok-4.3
The pith
Dirichlet L-functions for even characters have their exceptional zero bounded farther from 1.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By deriving a stronger explicit bound on L'(σ; χ) near σ = 1 from an average over characters combined with a lower bound at s = 1 and direct computation, the paper shows that any exceptional zero β belonging to an even character satisfies a stricter inequality β < 1 − c / log q than was previously available.
What carries the argument
An explicit estimate for L'(σ; χ) when σ is close to 1, constructed from an average result on Dirichlet characters together with a lower bound for L(1; χ) and computational checks.
If this is right
- The zero-free region for L(s, χ) with even χ is enlarged by a larger explicit constant.
- Results in prime-number theory that rely on zero-free regions for even characters receive quantitatively stronger explicit versions.
- The method treats even characters separately from odd ones, allowing the bound to be optimized for the even case alone.
Where Pith is reading between the lines
- The same averaging-plus-lower-bound approach could be rerun with any future improvement in the average estimate to push the constant c still higher.
- Extending the computational verification range to larger conductors would immediately tighten the bound for all even characters up to that size.
- Quantitative gains from the improved region could appear in explicit versions of the class-number problem or Linnik-type theorems that involve even characters.
Load-bearing premise
The claimed improvement depends on the accuracy of the explicit estimate for L'(σ; χ) near σ = 1 that is assembled from an average result over characters, a lower bound on L(1; χ), and computational verification; any undetected error in these three ingredients collapses the final bound on the zero.
What would settle it
An independent computation that finds either an even-character exceptional zero lying to the right of the new claimed bound or an L' value exceeding the asserted estimate near s = 1 would falsify the improvement.
read the original abstract
This paper improves the upper bound for the exceptional zeroes of Dirichlet L-functions with even characters. The result is obtained by improving on explicit estimate for $L'(\sigma;\chi)$ for $\sigma$ close to unity, using a result on the average of Dirichlet characters, and on the lower bound for $L(1;\chi)$, with computational aid.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to improve the upper bound on exceptional zeros of Dirichlet L-functions for even characters. The improvement is obtained by deriving a sharper explicit estimate for L'(σ; χ) when σ is close to 1, by combining an average result over Dirichlet characters, a lower bound for L(1; χ), and computational verification.
Significance. If the claimed bound holds with the stated constants, it would tighten the zero-free region for even characters and strengthen explicit versions of results on primes in arithmetic progressions or related arithmetic applications. The approach of blending analytic averages with computation is standard in this area, but the final numerical improvement rests entirely on the accuracy of the three external ingredients.
major comments (1)
- The central claim (improved zero bound) is load-bearing on the new explicit estimate |L'(σ; χ)| ≤ C(1-σ)^{-1} (or similar) for σ near 1. This estimate is assembled from an average over characters, a lower bound on L(1; χ), and a computational check; any looseness in the constants of these three ingredients directly invalidates the final zero-free region. The manuscript provides no independent cross-check, machine verification, or sensitivity analysis for the combined constants.
minor comments (1)
- [Abstract] The abstract does not state the numerical value of the improved bound or the explicit constant C achieved, which would help readers assess the size of the improvement.
Simulated Author's Rebuttal
We thank the referee for their comments on our manuscript. We address the single major comment below, providing clarification on the explicit nature of our estimates.
read point-by-point responses
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Referee: The central claim (improved zero bound) is load-bearing on the new explicit estimate |L'(σ; χ)| ≤ C(1-σ)^{-1} (or similar) for σ near 1. This estimate is assembled from an average over characters, a lower bound on L(1; χ), and a computational check; any looseness in the constants of these three ingredients directly invalidates the final zero-free region. The manuscript provides no independent cross-check, machine verification, or sensitivity analysis for the combined constants.
Authors: The explicit upper bound on |L'(σ; χ)| is assembled rigorously from three ingredients whose constants are stated explicitly in the paper: the average result over characters (Theorem 2.3, with its explicit constant), the lower bound on L(1; χ) (invoked from our prior work with its explicit constant), and the computational verification (detailed in Section 4 with explicit ranges for σ, character moduli, and error tolerances). The final constant C in the estimate is chosen strictly larger than the sum of the three contributions, incorporating a deliberate margin. Because every step is deterministic and fully explicit, the combined bound holds without requiring separate sensitivity analysis; the proofs themselves constitute the verification. We therefore see no need to alter the manuscript on this point. revision: no
Circularity Check
No circularity; derivation uses independent external inputs
full rationale
The paper's improvement on the bound for exceptional zeros is obtained by deriving an explicit estimate for L'(σ;χ) near σ=1 from an average result over Dirichlet characters, a lower bound for L(1;χ), and computational verification. These three ingredients are external to the present derivation and are not shown to reduce to any fitted parameter, self-definition, or self-citation chain within the paper itself. No equation or step equates the final zero bound to an input by construction, and the central claim retains independent content from the cited averages and bounds.
discussion (0)
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