The Goldbach exceptional set: an exponent 7/12 and conditional refinement

Ingo Alth\"ofer; J\'anos Pintz

REVIEW

The Goldbach exceptional set: an exponent 7/12 and conditional refinement

János Pintz, Ingo Althöfer

Number Theory · Submitted 11 Oct 2026 · 13 references

Pith's verdict: Conditional 7/12 Goldbach bound that is honestly conditional—clear dependencies, novel combination, but rests on two unverified preprints.

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arXiv ID2610.15559 (version 1)

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CitationIngo Alth\"ofer, J\'anos Pintz. "The Goldbach exceptional set: an exponent 7/12 and conditional refinement." arXiv:2610.15559 (2026). https://pith.science/paper/K3WGZF3T

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Editor's note

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This paper gives a conditional bound E(X) << X^{7/12+ε} under the assumption that all Dirichlet L-functions share a fixed zero-free half-plane (θ_0<1). The exponent 7/12 is new under that hypothesis, improving the previous conditional bound 0.709. The paper also proves a refinement (Theorem 3) that, under the stronger hypothesis H_θ (all zeros ≤ θ, 1/2 ≤ θ ≤ 2/3), yields exponents arbitrarily close to 1/2.

The novel ingredient is combining Pintz's major-arc formula with the exponential-sum estimate of Maynard–Pandey–Radziwiłł (MPR) from August 2026. On the minor arcs, the MPR estimate replaces the classical 4/5 exponent with 19/24, which directly tightens the exceptional-set exponent from 3/5 to 7/12. The conditional refinement (Section 4) is independent: it uses classical zero-density estimates and the stronger zero hypothesis to push the exponent further. That part is logically self-contained and shows the method's flexibility. The paper is fully transparent about its dependencies: the main theorem relies on the MPR estimate [3] and on the OpenAI 7/8 theorem [6], both recent preprints the authors do not verify. They state this explicitly, and the numerical chain [10] is mentioned for context only.

The soft spots are proportionate. The central result is completely conditional on [3] and [6]. The stress-test note is correct: if [3] is flawed or its uniformity fails, the 7/12 exponent collapses. The paper acknowledges this, so it's not a hidden flaw, but it is a load-bearing dependency. The zero-free half-plane hypothesis itself is extremely strong—far beyond any known result—so the practical reach is limited. The corollary invoking the OpenAI 7/8 theorem is fragile, as that result hasn't been independently confirmed. Theorem 3 avoids [3] and [6], but depends on the classical zero-density estimate (4.1) and the unproven H_θ. That part is well argued within its own frame.

This paper is for readers interested in the conditional side of the Goldbach exceptional set. It cleanly separates assumptions and uses a new exponential-sum estimate to get a better exponent under an optimistic hypothesis. The referee should check the application of [3] carefully. The paper deserves a serious referee: the mathematics is coherent, the novel combination is legitimate, and the transparency is high. I'd bring it to a reading group focused on analytic number theory or conditional results.

Referee report

Major objections (3)

[Theorem 1 and §2.2] The proof of Theorem 1 depends critically on the exponential-sum estimate [3, Theorem 1.1], reproduced as (2.5). That reference is an unpublished preprint (arXiv:2608.14777v1, August 2026) that has not been independently verified. The exponent 7/12 is directly tied to the 19/24 exponent in (2.6); if [3] is incorrect or its uniformity fails, the central result is unsupported. The paper should state Theorem 1 as conditional on [3] (not merely on H_θ0) or provide a self-contained proof of the needed estimate.
[Corollary 2] Corollary 2 invokes the 'uniform 7/8 theorem' [6], another non-peer-reviewed OpenAI preprint. The corollary is entirely conditional on that external input. The text should explicitly state 'If [6, Theorem 1.1] holds, then …' to avoid giving the impression of an unconditional result.
[§6 (AI-use declaration), final paragraph] The declaration states that 'this working draft does not represent their final approval of every statement.' This sentence casts doubt on whether the manuscript is ready for publication. The authors should either remove this disclaimer or clarify that the submitted version reflects their final, approved content. The authors acknowledge this: “The human authors retain responsibility for the final mathematical content, references, and submission, and this working draft does not represent their final approval of every statement.”

Minor objections (7)

[Abstract and §1] The function g(θ) is written as θ(5−4θ)/(2(2−θ)) but also as 2θ−3/2+3(1−θ)/(2−θ). The two forms are consistent; consider using the compact form throughout to avoid confusion.
[§2, after (2.1)] The line 'For fixed sufficiently large H, if no primitive Dirichlet L-function … has a zero in Res ≥ 1−H/log X, |Im s| ≤ √X' should specify that H is a constant to be chosen later; the dependence on H in (2.2) is via e^{-c_δ H} and the condition (2.3).
[§2.1, last line before §2.2] The phrase 'This excludes all the generalized exceptional zeros relevant to the formula, not only a possible real Siegel zero' could be expanded: a brief reminder of what 'generalized exceptional zeros' are would help readers.
[§4.2, equation (4.4)] The definition of a(θ) appears twice (once as (4.4) and later as h(θ) in (4.10) discussion). Consider unifying the notation to reduce redundancy.
[§4.2, proof of Lemma 4] The phrase 'the possible simple or double pole of the integrand at zero' is vague. A short explanation of how residues at s=0 are handled (or a reference to a standard treatment) would improve clarity.
[§5] The comparison with the Guth–Maynard density estimate is useful but could be condensed; the main point (that it does not improve the range 1/2 ≤ σ ≤ 2/3) is clear.
[General] There are minor typos: 'a fixed zero-free half-plane' should be 'a fixed zero-free half-plane' (repeated); 'conductor one' appears twice; line 'Since2P 2<X' needs a space. A careful proofreading is recommended.

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The paper

The paper claims

In the presence of a common zero-free half-plane for Dirichlet L-functions, the Goldbach exceptional set is bounded by X^{7/12+ε}, and under stronger zero hypotheses the exponent can be pushed strictly below that value.

The reading

The paper studies the Goldbach exceptional set $E(X)$, the number of even integers up to $X$ that cannot be written as a sum of two primes. Assuming that all Dirichlet $L$-functions share a fixed zero-free half-plane $\operatorname{Re} s > \theta_0$ for some $\theta_0 < 1$, it proves $E(X) \ll_\varepsilon X^{7/12+\varepsilon}$. A corollary invokes an external theorem that supplies $\theta_0 = 7/8$ to make the bound unconditional up to that assumption. Under the stronger hypothesis $H_\theta$ (every nontrivial Dirichlet zero has real part at most $\theta$, with $\frac12\le\theta\le\frac23$), it obtains $E(X) \ll_{\theta,\varepsilon} X^{g(\theta)+\varepsilon}$, where $g(\theta)=\theta(5-4\theta)/(2(2-\theta))$ is strictly less than $7/12$ for any fixed $\theta<2/3$. All results are conditional on unproven properties of Dirichlet $L$-function zeros.

What carries the argument

For Theorem~1, the main tools are Pintz's explicit formula [8] for the major arcs and the Maynard–Pandey–Radziwiłł exponential‑sum bound [3] which gives a $19/24$ exponent on $X$ in the minor‑arc supremum, leading to $7/12$ after Bessel's inequality. For Theorem~3, the central object is Lemma~4, an asymptotic expansion for the smoothed exponential sum $F_X(\alpha)$ near rational $a/q$. It uses character expansion, contour integration, and the fixed‑modulus zero‑density estimate $N_q(\sigma,U) \ll (qU)^{3(1-\sigma)/(2-\sigma)}\log^9(qU)$ (from [11] or [4]) to bound the contribution of zeros. The resulting uniform approximation yields a bound $X^{a(\theta)+\eta}$ for the $L^\infty$ deviation, which through $L^2$ comparison gives the exceptional‑set exponent $g(\theta)=2a(\theta)-1$.

Extended reading notes

Core claim

The central discovery is that the exponent in the Goldbach exceptional set can be conditionally reduced to $7/12$ (Theorem~1) and, under the uniform zero bound $H_\theta$ with $\theta\le2/3$, to the even smaller exponent $g(\theta)=\theta(5-4\theta)/(2(2-\theta))$ (Theorem~3). The proof of Theorem~1 combines Pintz's explicit major‑arc formula with the exponential‑sum estimate of Maynard, Pandey, and Radziwiłł to handle the minor arcs, while Theorem~3 uses a classical fixed‑modulus zero‑density estimate in place of the new exponential‑sum bound. The key insight is that once the major arcs contribute a positive main term (guaranteed by the zero‑free hypothesis), the minor‑arc estimate becomes the limiting factor, and the choice of zero‑density estimate determines the final exponent.

Load-bearing premise

Both theorems assume unproven properties of Dirichlet $L$-function zeros: Theorem~1 requires a common zero‑free half‑plane (supplied by an external, non‑trivial theorem), and Theorem~3 requires that all nontrivial zeros have real part at most $\theta$ for some fixed $\theta\le 2/3$.

Editorial extensions

If this is right

If the common zero‑free half‑plane assumption holds (e.g. as claimed by an external theorem), the Goldbach exceptional set has size at most $X^{7/12+\varepsilon}$, improving on the previous best exponent $0.684$ from unpublished numerical work.
For any fixed $\theta<2/3$ for which $H_\theta$ holds, the exponent drops below $7/12$; for example $H_{5/8}$ gives $E(X)\ll X^{25/44+\varepsilon}$, a gap of $1/66$.
The proof of Theorem~3 is independent of the new exponential‑sum theorem and relies only on classical zero‑density estimates, showing that the conditional improvement is robust to the choice of minor‑arc technology.
The method makes explicit how the location of Dirichlet zeros quantitatively controls the number of even numbers that fail to be sums of two primes.

Reading between the lines

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

The entire edifice of Theorem~1 and its corollary rests on an external, non‑trivial zero‑free theorem that has not yet appeared in peer‑reviewed literature. Until that result is independently verified, the $7/12$ exponent remains a conditional implication, not a proven bound.
Theorem~3 shows that if one is willing to assume $H_\theta$ with $\theta$ as high as $2/3$, the exponent $7/12$ is already the worst case; any improvement on the zero hypothesis immediately pays off in the exceptional set. This suggests that further progress on the Goldbach problem may come from better zero‑free regions for Dirichlet $L$-functions.
The paper explicitly notes that the Guth–Maynard zero‑density estimate does not sharpen the argument because its range of applicability ($\sigma>7/10$) lies above the $\theta\le2/3$ considered here. A fixed‑modulus analogue of that estimate would not help either, as the exponent $D(\sigma)$ in (4.1) is already optimal for $\sigma\le2/3$ among known estimates.

Original abstract

Let $E(X)$ count the even integers between $4$ and $X$ that are not sums of two primes. We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwi{\l}{\l} to prove $E(X)\ll_\varepsilon X^{7/12+\varepsilon}$ whenever all Dirichlet $L$-functions have a common fixed zero-free half-plane $\Rea s>\theta_0$, with $\theta_0<1$. The uniform $7/8$ theorem reported in the OpenAI mathematics release supplies this input. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most $\theta$, where $1/2\leq\theta\leq2/3$, we prove \[ E(X)\ll_{\theta,\varepsilon}X^{g(\theta)+\varepsilon}, \qquad g(\theta)=\frac{\theta(5-4\theta)}{2(2-\theta)}. \] Thus every fixed $\theta<2/3$ gives an exponent strictly below $7/12$. The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.

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.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-14: 1 request, 5,242 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.79 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.73 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Let $E(X)$ count the even integers between $4$ and $X$ that are not sums of two primes. We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwi{\l}{\l} to prove $E(X)\ll_\varepsilon X^{7/12+\varepsilon}$ whenever all Dirichlet $L$-functions have a common fixed zero-free half-plane $\Rea s>\theta_0$, with $\theta_0<1$. The uniform $7/8$ theorem reported in the OpenAI mathematics release supplies this input. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most $\theta$, where $1/2\leq\theta\leq2/3$, we prove \[ E(X)\ll_{\theta,\varepsilon}X^{g(\theta)+\varepsilon}, \qquad g(\theta)=\frac{\theta(5-4\theta)}{2(2-\theta)}. \] Thus every fixed $\theta<2/3$ gives an exponent strictly below $7/12$. The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $E(X)$ count the even integers between $4$ and $X$ that are not sums of two primes.	background 1.00	0.18	0.02	asserted 0.64
s1	We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwi{\l}{\l} to prove $E(X)\ll_\varepsilon X^{7/12+\varepsilon}$ whenever all Dirichlet $L$-functions have a common fixed zero-free half-plane $\Rea s>\theta_0$, with $\theta_0<1$.	result 0.71	0.92	0.70	asserted 1.00
s2	The uniform $7/8$ theorem reported in the OpenAI mathematics release supplies this input.	method 0.49	0.05	0.04	asserted 1.00
s3	Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most $\theta$, where $1/2\leq\theta\leq2/3$, we prove \[ E(X)\ll_{\theta,\varepsilon}X^{g(\theta)+\varepsilon}, \qquad g(\theta)=\frac{\theta(5-4\theta)}{2(2-\theta)}.	result 0.98	0.92	0.31	asserted 1.00
s4	\] Thus every fixed $\theta<2/3$ gives an exponent strictly below $7/12$.	interpretation 0.91	0.78	0.18	asserted 1.00
s5	The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.	method 0.79	0.60	0.03	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (13)

[3]

J. Maynard, M. Pandey, and M. Radziwiłł,Exponential sums over primes, arXiv:2608.14777v1 (2026), 14 August 2026. arXiv version

View on Pith Pith review arXiv 2026
[6]

OpenAI,The Quasi-Riemann Hypothesis: A Zero-Free Half-PlaneRes>7/8, OpenAI Math Release preprint, 30 September 2026, family 003. Theorem 1.1. Archived repository version at commitfd4aeeb2ee4f, consulted 10 October 2026

View on Pith 2026
[10]

J. Pintz and I. Althöfer, with ChatGPT assistance,Goldbach exceptional-set calculations, unpublished research manuscript and certificate chain, September 2026. The intermediate addendum goldbach_068601_addendum.tex is dated 12 September 2026. Later research notes, includingGoldbach exponents below0.69: a comparison with Pintz’s 2018 method andExtrapolatin...

View on Pith 2026
[1]

L. Guth and J. Maynard,New large value estimates for Dirichlet polynomials, arXiv:2405.20552v2 (2024). arXiv version

View on Pith Pith review arXiv 2024
[2]

D. R. Heath-Brown,Zero-free regions for DirichletL-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3)64(1992), no. 2, 265–338

View on Pith 1992
[4]

H. L. Montgomery,Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, vol. 227, Springer-Verlag, Berlin–New York, 1971

View on Pith 1971
[5]

H. L. Montgomery and R. C. Vaughan,The exceptional set in Goldbach’s problem, Acta Arithmetica27(1975), 353–370

View on Pith 1975
[7]

OpenAI,The quasi-Riemann hypothesis: scope of the Lean formalization, OpenAI Math Release, family 003. Formalization scope note at the same repository commit, consulted 10 October 2026

View on Pith 2026
[8]

J. Pintz,A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach problem and the Generalized Twin Prime Problem, Acta Arithmetica210(2023), 53–94. doi:10.4064/aa220728-31-3. Preprint: arXiv:1804.05561v1 (2018); theorem and equation numbers referring to this preprint are explicitly identified in the text

View on Pith Pith review arXiv 2023
[9]

J. Pintz,A new explicit formula in the additive theory of primes with applications II. The exceptional set in Goldbach’s problem, arXiv:1804.09084v2 (2018)

View on Pith Pith review arXiv 2018
[11]

D. Wolke,Primes with preassigned digits, Acta Arithmetica119(2005), no. 2, 201–209. See equation (2.2.5), p. 203. Publisher’s PDF

View on Pith 2005
[12]

T. Xylouris,On Linnik’s constant, arXiv:0906.2749 (2009)

View on Pith Pith review arXiv 2009
[13]

G. Zhao,The exceptional set of Goldbach problem, arXiv:2511.05631v1 (2025). János Pintz HUN-REN Alfréd Rényi Institute of Mathematics Reáltanoda u. 13–15, H-1053 Budapest, Hungary E-mail:pintz@renyi.hu 11 Ingo Althöfer Friedrich-Schiller University Jena Faculty of Mathematics and Computer Science D-07740 Jena, Germany E-mail:ingo.althoefer@uni-jena.de 12

View on Pith arXiv 2025

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes
math.NT 2026-10 conditional novelty 7.0 of 10

Under a quasi-Riemann hypothesis with zero real parts at most Theta < 11/12, the exceptional sets in the binary Goldbach and prime-difference problems are O(N^{1/2} (log N)^{100}).
The exceptional set of Goldbach numbers among polynomial values
math.NT 2026-11 conditional novelty 6.0 of 10

For every fixed integer polynomial phi of degree k, all but N^{alpha_k+epsilon} of the values 2phi(n), n<=N, are Goldbach sums, with alpha_2=1/2, alpha_3=5/8, alpha_4=3/4, and alpha_k=1-1/(k+1).

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [3]

J. Maynard, M. Pandey, and M. Radziwiłł,Exponential sums over primes, arXiv:2608.14777v1 (2026), 14 August 2026. arXiv version

View on Pith Pith review arXiv 2026

[2] [6]

OpenAI,The Quasi-Riemann Hypothesis: A Zero-Free Half-PlaneRes>7/8, OpenAI Math Release preprint, 30 September 2026, family 003. Theorem 1.1. Archived repository version at commitfd4aeeb2ee4f, consulted 10 October 2026

View on Pith 2026

[3] [10]

J. Pintz and I. Althöfer, with ChatGPT assistance,Goldbach exceptional-set calculations, unpublished research manuscript and certificate chain, September 2026. The intermediate addendum goldbach_068601_addendum.tex is dated 12 September 2026. Later research notes, includingGoldbach exponents below0.69: a comparison with Pintz’s 2018 method andExtrapolatin...

View on Pith 2026

[4] [1]

L. Guth and J. Maynard,New large value estimates for Dirichlet polynomials, arXiv:2405.20552v2 (2024). arXiv version

View on Pith Pith review arXiv 2024

[5] [2]

D. R. Heath-Brown,Zero-free regions for DirichletL-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3)64(1992), no. 2, 265–338

View on Pith 1992

[6] [4]

H. L. Montgomery,Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, vol. 227, Springer-Verlag, Berlin–New York, 1971

View on Pith 1971

[7] [5]

H. L. Montgomery and R. C. Vaughan,The exceptional set in Goldbach’s problem, Acta Arithmetica27(1975), 353–370

View on Pith 1975

[8] [7]

OpenAI,The quasi-Riemann hypothesis: scope of the Lean formalization, OpenAI Math Release, family 003. Formalization scope note at the same repository commit, consulted 10 October 2026

View on Pith 2026

[9] [8]

J. Pintz,A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach problem and the Generalized Twin Prime Problem, Acta Arithmetica210(2023), 53–94. doi:10.4064/aa220728-31-3. Preprint: arXiv:1804.05561v1 (2018); theorem and equation numbers referring to this preprint are explicitly identified in the text

View on Pith Pith review arXiv 2023

[10] [9]

J. Pintz,A new explicit formula in the additive theory of primes with applications II. The exceptional set in Goldbach’s problem, arXiv:1804.09084v2 (2018)

View on Pith Pith review arXiv 2018

[11] [11]

D. Wolke,Primes with preassigned digits, Acta Arithmetica119(2005), no. 2, 201–209. See equation (2.2.5), p. 203. Publisher’s PDF

View on Pith 2005

[12] [12]

T. Xylouris,On Linnik’s constant, arXiv:0906.2749 (2009)

View on Pith Pith review arXiv 2009

[13] [13]

G. Zhao,The exceptional set of Goldbach problem, arXiv:2511.05631v1 (2025). János Pintz HUN-REN Alfréd Rényi Institute of Mathematics Reáltanoda u. 13–15, H-1053 Budapest, Hungary E-mail:pintz@renyi.hu 11 Ingo Althöfer Friedrich-Schiller University Jena Faculty of Mathematics and Computer Science D-07740 Jena, Germany E-mail:ingo.althoefer@uni-jena.de 12

View on Pith arXiv 2025

#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let \(E(X)\) count the even integers between \(4\) and \(X\) that are not sums of two primes.	background 1.00	0.18	0.02	asserted 0.64
s1	We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwi{\l}{\l} to prove \(E(X)\ll_\varepsilon X^{7/12+\varepsilon}\) whenever all Dirichlet \(L\)-functions have a common fixed zero-free half-plane \(\Rea s>\theta_0\), with \(\theta_0<1\).	result 0.71	0.92	0.70	asserted 1.00
s2	The uniform \(7/8\) theorem reported in the OpenAI mathematics release supplies this input.	method 0.49	0.05	0.04	asserted 1.00
s3	Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most \(\theta\), where \(1/2\leq\theta\leq2/3\), we prove \[ E(X)\ll_{\theta,\varepsilon}X^{g(\theta)+\varepsilon}, \qquad g(\theta)=\frac{\theta(5-4\theta)}{2(2-\theta)}.	result 0.98	0.92	0.31	asserted 1.00
s4	\] Thus every fixed \(\theta<2/3\) gives an exponent strictly below \(7/12\).	interpretation 0.91	0.78	0.18	asserted 1.00
s5	The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.	method 0.79	0.60	0.03	asserted 1.00