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An explicit version of Carlson's theorem

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims an explicit Carlson-type zero-density bound: $N(\sigma,T) \le K(\sigma,T_0)T^{4\sigma(1-\sigma)}(\log T)^{5-2\sigma}$ for $T\ge T_0\ge 3\cdot 10^{12}$ and $\sigma\ge 0.6$.

desk verdict The paper advertises a sharper log-exponent than it proves; the proof ends with Carlson's original log^4, so the main claim as stated collapses. read the letter →

arxiv 2412.02068 v1 pith:BEM3ZRCZ submitted 2024-12-03 math.NT

Shashi Chourasiya This is my paper · ORCID
classification math.NT MSC 11N5611N3711M06
keywords zerodensityestimatesRiemannzetafunctionCarlson'stheoremexplicitmollifiedapproximatefunctionalequationLittlewood'smethodnontrivialzeros
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to turn Carlson's classical zero-density estimate for the Riemann zeta function into an explicit, usable inequality. It claims that for $T \ge T_0 \ge 3\cdot 10^{12}$ and $\sigma \ge 0.6$, the number $N(\sigma,T)$ of nontrivial zeros with real part greater than $\sigma$ and imaginary part up to $T$ satisfies $N(\sigma,T) \le K(\sigma,T_0)\,T^{4\sigma(1-\sigma)}(\log T)^{5-2\sigma}$, with numerical constants $K(\sigma,T_0)$ tabulated in the paper. This would be the first explicit version of Carlson's bound, and the logarithmic exponent $5-2\sigma$ is meant to sharpen Carlson's asymptotic $\log^4 T$. Zero-density bounds of this kind control how many zeros can lie off the critical line, which matters for results on primes in short intervals and related arithmetic questions. The proof uses Littlewood's zero-counting lemma on a mollified zeta function and bounds the resulting integrals explicitly.

What carries the argument

The central object is the mollified zeta function $h_X(s)=\zeta(s)M_X(s)(2-\zeta(s)M_X(s))$, where $M_X(s)=\sum_{n\le X}\mu(n)n^{-s}$ is the Möbius mollifier and $X=T^{2\sigma-1}\log T$. Zeros of $\zeta$ are zeros of $h_X$, so Littlewood's rectangle lemma converts $N(\sigma,T)$ into integrals of $\log|h_X|$ and $\arg h_X$. The main estimate is a second-moment bound $\int_H^T |f_X(\alpha+it)|^2\,dt \le C_3(\sigma,T_0)\,T^{4\sigma(1-\sigma)}\log^3 T$, where $f_X=\zeta M_X-1$; it is obtained from an explicit approximate functional equation for $\zeta$ and a mean-value theorem for Dirichlet polynomials, and the remaining argument and vertical-log integrals are handled by two cited lemmas.

What would settle it

Check the hypotheses of the two quoted lemmas for $h_X(s)=\zeta(s)M_X(s)(2-\zeta(s)M_X(s))$ with $X=T^{2\sigma-1}\log T$ and recompute the final display in Section 5: if the chain yields $\log^4 T$ rather than $(\log T)^{5-2\sigma}$, the theorem as stated is not established by the given proof.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that Carlson's asymptotic estimate $N(\sigma,T)=O(T^{4\sigma(1-\sigma)}\log^4 T)$ can be made fully explicit with a reasonably small constant. Theorem 1.1 asserts that for $\sigma\ge 0.6$ and $T\ge T_0\ge 3\cdot 10^{12}$, one has $N(\sigma,T) \le K(\sigma,T_0)\,T^{4\sigma(1-\sigma)}(\log T)^{5-2\sigma}$, with $K(\sigma,T_0)$ defined in (5.3), tabulated for several $T_0$ values, and tending to $\frac{1}{2\pi}\frac{0.68\,\sigma(2\sigma-1)^2}{1-0.54\,\sigma(1-\sigma)}$ as $T_0$ grows. The author presents the result as an improvement in the exponent of the logarithm factor over Carlson's original bound and compares it numerically with the best existing explicit estimates, identifying the range of $\sigma$ for which the new inequality is sharpest. The intended contribution is a small explicit constant and a logarithmic-power refinement obtained from relatively elementary ingredients: an approximate functional equation, a mean-value estimate for Dirichlet polynomials, and Littlewood's counting argument.

Load-bearing premise

The estimate collapses if the quoted bounds on the argument and vertical-logarithm integrals do not apply to the specific mollified function $h_X(s)=\zeta(s)M_X(s)(2-\zeta(s)M_X(s))$ with $X=T^{2\sigma-1}\log T$, and the paper does not verify the stated hypotheses of those quoted lemmas for that function.

Editorial extensions

If this is right

  • For $\sigma\in[0.6,2/3]$ and sufficiently large $T$, the new bound becomes sharper than the best near-critical explicit estimate; the paper gives crossover points such as $T\ge 9.48\cdot 10^{308}$ for $T_0=3\cdot 10^{12}$ and $T\ge 1.43\cdot 10^{236}$ for $T_0=10^{200}$.
  • The constants $K(\sigma,T_0)$ in Table 1 are ready for direct use in applications, and their large-$T_0$ limit has a simple closed form.
  • Because the proof relies only on the approximate functional equation, mean-value estimates for Dirichlet polynomials, and Littlewood's counting lemma, the same scheme is designed to transfer to Dirichlet $L$-functions and Dedekind zeta functions.
  • The estimate is strongest for $\sigma\le 2/3$, so it fills a region of the $(\sigma,T)$-plane where earlier explicit bounds are weaker.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A close reading of Section 5 shows that the displayed chain of inequalities ends with a factor $\log^4 T$, whereas the theorem statement and abstract advertise $(\log T)^{5-2\sigma}$; for $\sigma>1/2$ these two exponents differ, so the advertised logarithmic improvement does not appear in the final displayed calculation. This is an editorial observation about the text, not a verdict on whether a m
  • If the logarithm exponent is taken to be $4$, the result remains an explicit Carlson-type estimate with explicit constants, and the crossover comparisons with other explicit estimates would need to be recomputed.
  • A finite test of the method would be to instantiate every displayed inequality in Section 5 for a fixed $\sigma$ (say $\sigma=0.6$) and check exactly which power of $\log T$ the chain yields.
  • The same mollifier construction with $X=T^{2\sigma-1}\log T$ could be tried for other $L$-functions, but portability requires proving the analogue of the two quoted argument and vertical-logarithm bounds for the new $L$-function.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper aims to give the first explicit version of Carlson's zero-density estimate for the Riemann zeta function, claiming N(σ,T) ≤ K(σ,T0) T^{4σ(1−σ)} (log T)^{5−2σ} for T ≥ T0 ≥ 3·10^12 and σ ≥ 0.6, with K defined in (5.3). The proof follows Carlson's method via Littlewood's counting lemma, a mollified zeta function, a second-moment estimate, and explicit bounds from Kadiri–Lumley–Ng. The advertised improvement over Carlson's log^4 T consists of replacing the logarithm exponent 4 by 5−2σ.

Significance. If correct, an explicit Carlson-type estimate with the improved logarithm exponent would be a useful addition to the explicit zero-density literature, complementing recent work by Kadiri–Lumley–Ng, Simonič, and Bellotti. The paper draws on published external lemmas and does not rely on curve fitting or circular reasoning, which is a strength. However, the central advertised exponent is not derived, and the transferred use of two key lemmas is not fully justified; as it stands, the main theorem is unsupported.

major comments (3)
  1. [§5, Theorem 1.1 and final display] The proof concludes with N(σ,T) ≤ K(σ,T0) T^{4σ(1−σ)} log^4 T, not the advertised log^{5−2σ} T. In the final step, the prefactor 1/(σ−α) equals log T, since α = σ − 1/log T, so the first integral contributes at most (C3(σ,T0)/(2π)) T^{4σ(1−σ)} log^4 T. There is no subsequent step that converts log^4 T into log^{5−2σ} T; indeed, for σ ≥ 0.6, 5−2σ < 4, so the claimed exponent is stronger and does not follow. The constant K(σ,T0) in (5.3) is explicitly written for the log^4 T bound, and the theorem and abstract reuse this K with log^{5−2σ} T, which is inconsistent. The main claim of the paper is therefore not established.
  2. [§4, text near (4.4)] The sentence 'Carlson obtained as 3' misstates the classical bound (1.2), which has log^4 T. Since the whole novelty of the paper is the improved logarithm exponent, this misstatement compounds the exponent error in Theorem 1.1 and should be corrected or removed.
  3. [§5, Lemmas 5.1 and 5.2] Lemmas 5.1 and 5.2 are imported from Kadiri–Lumley–Ng without a verification that their hypotheses hold for the particular h_X(s) = ζ(s)M_X(s)(2 − ζ(s)M_X(s)) with X = T^{2σ−1} log T. The statement of Lemma 5.1 contains 'on fixing some parameters' without specifying those choices, and the paper does not show that the required conditions (e.g., the ranges of α, β, η, and the size of X) are satisfied in the present setting. If these bounds do not transfer to this mollifier, the estimates for the argument and vertical-boundary integrals in (2.4) fail. This gap is independent of the exponent mismatch and also affects the final bound.
minor comments (3)
  1. [Abstract and Theorem 1.1] The abstract states a constant 0.78, but Theorem 1.1 gives K(σ,T0) and Table 1 lists several different values; the relationship between 0.78 and K(σ,T0) should be clarified.
  2. [Throughout] There are several typographical issues, including 'seperately' in §4, 'integals' in §5, and inconsistent notation for Simonič's name and reference formatting. These do not affect the mathematics but should be corrected.
  3. [§4, equation (4.5) and (4.6)] The dyadic summation bound leading to the factor 1/(1 − 0.54σ(1−σ)) is plausible, but the origin of the constant 0.54 should be stated explicitly, since it appears to be an input to the summation and is not derived in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation rests on independently proved external lemmas; the theorem/proof exponent mismatch is an internal correctness issue, not a circular one.

full rationale

The derivation chain for Theorem 1.1 is not circular. The zero-counting inequality (2.4) is a standard Littlewood argument, and the second-moment bound (4.5)-(4.6) is obtained from explicit estimates proved in this paper using Lemma 3.1 (from Cully-Hugill and Trudgian [CHT21]), Lemma 3.2 (from Simonič [Sim19]), and a Montgomery-Vaughan mean value theorem in the form derived by Ramaré [Ram15]. The argument and log|h_X| bounds are imported from Lemmas 5.1 and 5.2 of Kadiri, Lumley, and Ng [KLN18]. None of these inputs is defined in terms of the target N(σ,T), and none is fitted to the theorem's claimed constants. The paper says it 'strongly relies' on Carlson's original method, but using the same method is not circular. Although [CHT21] and [PT20] involve the author's supervisor, T. Trudgian, the author of the present paper is not an author of those works, and those results are published with independent proofs rather than being asserted uniqueness theorems. The serious defect in the paper is that Section 5 concludes with N(σ,T) ≤ K(σ,T0) T^{4σ(1−σ)} log^4 T, while Theorem 1.1 and the abstract advertise log^{5−2σ} T; no displayed step converts log^4 into log^{5−2σ}, and for σ > 1/2 the advertised exponent is strictly stronger. That is an internal inconsistency or correctness failure, not a reduction of a prediction to a fitted input, so it does not raise the circularity score. Similarly, the apparent lack of hypothesis checks in transferring [KLN18] Lemmas 5.1 and 5.2 to this particular mollifier is a rigor gap, not circularity. Therefore the circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The proof is entirely a chain of explicit estimates from the literature; the load-bearing external inputs are the divisor-sum bound, the approximate functional equation, the mean value theorem, and the Kadiri-Lumley-Ng argument bounds. The paper contributes the combination and the constant tracking, but does not prove the KLN lemmas and does not check their hypotheses.

free parameters (1)
  • dyadic-sum ratio bound 0.54 = 0.54 (upper bound for (1/2)^{4σ(1-σ)} on σ∈[0.6,2/3])
    Chosen by hand in section 4 to bound the geometric series when passing from (4.5) to (4.6); the notation is unclear and likely should be a power.
assumptions (5)
  • domain assumption Riemann hypothesis verified for T ≤ 3·10^12 (Platt-Trudgian)
    Used in section 1 to restrict T ≥ T0 ≥ 3·10^12.
  • standard math Divisor sum bound ∑_{n≤t} d(n)^2 ≤ (1/4) t log^3 t for t ≥ 433 (Cully-Hugill-Trudgian)
    Used in Lemma 3.1 to derive (3.3).
  • standard math Approximate functional equation (3.5) with error 1.755 t^{-σ} (Simonič)
    Used in section 4 to decompose f_X(s) into A and B.
  • standard math Montgomery-Vaughan mean value theorem in Ramaré's form (3.7)
    Core mean value estimate used in Lemma 3.3.
  • domain assumption Argument and log|h_X| bounds from Kadiri-Lumley-Ng (Lemmas 5.1 and 5.2)
    Imported without full proof; their hypotheses are not checked for the specific mollifier h_X(s) = ζ(s)M_X(s)(2 − ζ(s)M_X(s)).

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Pith. "Pith review of An explicit version of Carlson's theorem." pith.science (2026). https://pith.science/paper/BEM3ZRCZ

@misc{pith2026241202068,
  author       = {Pith},
  title        = {Pith review of: An explicit version of Carlson's theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BEM3ZRCZ}},
  note         = {Machine review of arXiv:2412.02068}
}
abstract

Let $N(\sigma,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $\sigma$ and imaginary part lying between $0$ and $T$. In this article, we provide an explicit version of Carlson's zero density estimate, that is, $N(\sigma, T) \leq 0.78 T^{4 \sigma (1- \sigma)} (\log T)^{5-2 \sigma} $, with a slight improvement in the exponent of the logarithm factor.

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Works this paper leans on

3 extracted references · 1 canonical work pages

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    arXiv 2023

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    Two explicit divisor sums

    [CHT21] M. Cully-Hugill and T. Trudgian. “Two explicit divisor sums”. In : The Ramanujan Journal (2021), pp. 1–9. [GM24] L. Guth and J. Maynard. “New large value estimates for Dirich let poly- nomials”. In: arXiv preprint arXiv:2405.20552 (2024). [Hux71] M. N. Huxley. “On the difference between consecutive prime s”. In: In- vent. Math. 15 (1971), pp. 164–1...

    arXiv 2021

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    On a density theorem of Yu. V. Linnik

    [Tur61] P. Tur` an. “On a density theorem of Yu. V. Linnik”. In: Magyar Tud. Akad. Mat. Kutat` o Int. K¨ azl.6 (1961), pp. 165–170. School of Science, The University of New South W ales, Canberra , Australia Email address : s.chourasiya@unsw.edu.au

    work page 1961

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