The exceptional set of Goldbach numbers among polynomial values

A.Perelli; D.Chen; I.Alth\"ofer; J.Pintz

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The exceptional set of Goldbach numbers among polynomial values

I.Althöfer, D.Chen, A.Perelli, J.Pintz

Number Theory · Submitted 6 Nov 2026 · 10 references

Pith's verdict: Clean, genuinely new exponents, but Theorem 1 is announced unconditionally while riding on theta<=7/8 from an unverified OpenAI preprint — a worthy referee job, provided the hypothesis is brought onto the theorem statement.

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arXiv ID2611.09386 (version 1)

Pith ID76SCQAKH, Pith's permanent record of this paper and its review

CitationA.Perelli, D.Chen, I.Alth\"ofer, J.Pintz. "The exceptional set of Goldbach numbers among polynomial values." arXiv:2611.09386 (2026). https://pith.science/paper/76SCQAKH

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Editor's note

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Short version: the internal argument is clean and the exponents are genuinely new, but Theorem 1 as stated is not what is proved. The proof depends on theta <= 7/8, quoted from an OpenAI preprint, a zero-free half-plane far beyond anything established, and the abstract and theorem state the result without that condition. The stress-test note holds up on reading the paper.

What's actually new: for every degree k>=2, explicit exceptional-set exponents alpha_2=1/2, alpha_3=5/8, alpha_4=3/4, alpha_k=1-1/(k+1), replacing Bruedern-Kawada-Wooley's non-explicit N^{1-c0/k}. The machinery checks out: the Holder step (4.6), the moment exponent (k+2b)/q, Lemma G's divisor argument for quadratics, and the dyadic summation. It is short, readable, and the optimization over q and b is the right way to see where the exponents come from. Lemma G is a nice little argument. Section 5 is honestly framed as conditional on assumption (5.1), and the closing remark that no explicit beta is known is the kind of candor you do not always get.

The soft spot is single and load-bearing: Theorem 1 and the abstract promise an unconditional theorem, but the proof enters Theorem B's hypothesis theta<11/12 only via Theorem A (theta<=7/8) from [OAI2026a]. That preprint is unpublished; the accompanying note is a formalization scope note, not a machine-checked proof; the paper itself reports no formal verification; and theta<=7/8 is dramatically stronger than known regions of the form 1-c/log(q(|t|+2)). If that claim fails, or is narrower than quoted, the main theorem has nothing under it. The authors do acknowledge the reliance in Section 2, so this is not hidden — but the paper should carry the condition on the theorem statement itself. Theorems B and C from Li-Liu are cited rather than proved, yet used inside their stated hypotheses; that is normal practice, and not where I would push.

Who gets value: number theorists working on the circle method, plus anyone tracking the OpenAI quasi-Riemann hypothesis claim, because this paper is now the sharpest example of the field's dependence on it. Referee it — the internal proof deserves scrutiny, and the exchange will force public evaluation of Theorem A. But I would require the theorem be restated with theta<11/12 made explicit, or the authors must clearly separate the conditional from the unconditional claim.

Referee report

Significance

If Theorems A, B and C are accepted, the result would improve the earlier N^{1−c_0/k} bound of Brüdern–Kawada–Wooley with explicit exponents that are < 1 for every degree, and the Hölder machinery is a clean and apparently correct way to exploit arbitrary-subset mean-value estimates. The proof of the internal reduction is easy to follow; the moment computations, the exponent (k+2b)/q, and the dyadic summation all check out, and I found no circularity. The almost-everywhere application in Section 5 is a nice corollary. However, the decisive input θ ≤ 7/8 is a fixed zero-free half-plane for all primitive Dirichlet L-functions, a statement far beyond current unconditional knowledge and supported here only by an unpublished source; the major-arc and minor-arc inputs B and C are likewise from an unpublished preprint. The paper's contribution is therefore best assessed as a conditional theorem with a clear reduction; as written, the unconditional framing is not supported.

Major objections (2)

[§2, Theorem A; §4, Eq. (4.6)] Theorem 1 and the abstract are announced unconditionally, but the proof reaches (4.6) only after importing Theorem A (θ ≤ 7/8) from the unpublished preprint [OAI2026a]. The paper gives no proof or summary of Theorem A, and [OAI2026b] is described only as a formalization scope note; a fixed zero-free half-plane Re s > 7/8 for all primitive Dirichlet L-functions is far stronger than current unconditional zero-free regions. The authors do acknowledge the reliance in Section 2, but the statement of Theorem 1 does not carry the qualification. Please restate the abstract and Theorem 1 as conditional on this input, or at minimum on the weaker hypothesis θ < 11/12, which is all the later argument needs, or provide a verifiable proof of the input.
[§2, Theorems B and C] The fourth-moment bound (2.1) and the major-arc asymptotic (2.2) are taken from [LL2026], also an unpublished preprint, and both are load-bearing: (2.2) supplies the positive lower bound (1.17), while (2.1) enters the Hölder step (4.6). The manuscript therefore currently establishes a reduction of Theorem 1 to three external results rather than a self-contained proof. I ask that the final theorem and Section 5 be explicitly labelled conditional on Theorems A, B and C, and that the exact statements and hypotheses be checked against [LL2026]; if the journal requires proofs of all external inputs, that requirement is not yet met.

Minor objections (4)

[§1, Eq. (1.4)] The same symbol E_{k,φ}(N) is used both for the exceptional set and for its cardinality; please use distinct notation, for example a calligraphic symbol for the set.
[§4, Eqs. (4.5)–(4.6)] The Hölder step is correct but terse; a sentence naming the exponents q, q/(q−2), q and the use of Parseval's identity on the L^2 factor would help readers verify the exponent L^{1+98/q}.
[§5, Proposition] The proposition is phrased under a general assumption (5.1), but the surrounding discussion applies it with c_k = α_k from Theorem 1; this identification should be stated explicitly, and the conditional status of Theorem 1 should be repeated in Section 5.
[§3, Lemma G proof] In the divisor-bound step, the phrase 'with a suitably smaller exponent' could be made explicit: for each fixed ε > 0 one obtains r(t) ≪_{ψ,ε} M^ε for t ≠ 0.

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The paper

The paper claims

All but $N^{\alpha_k+o(1)}$ of the numbers $2\phi(n)$, $n\le N$, are sums of two primes, with $\alpha_2=1/2$, $\alpha_3=5/8$, $\alpha_4=3/4$, $\alpha_k=1-1/(k+1)$ for $k>4$, conditional on a fixed zero-free half-plane for Dirichlet…

The reading

Goldbach's conjecture says every even number above 2 is a sum of two primes; this paper asks how often the special even numbers $2\phi(n)$, where $\phi$ is a fixed integer polynomial of degree $k\ge 2$, escape being such a sum. The answer claimed is that the failures are rare: for $n\le N$ the exceptional indices number at most $N^{1/2+\varepsilon}$ when $k=2$, $N^{5/8+\varepsilon}$ when $k=3$, $N^{3/4+\varepsilon}$ when $k=4$, and $N^{1-1/(k+1)+\varepsilon}$ for $k>4$. The saving is meaningful because the values have size about $N^k$: the best known bound for the classical Goldbach exceptional set, $E(X)\ll X^{1/2+\varepsilon}$, would only give about $N^{k/2}\ge N$ exceptions here, which is no saving at all. The price is a strong hypothesis the paper states plainly: the proof assumes all primitive Dirichlet $L$-functions are zero-free in the fixed half-plane $\operatorname{Re}s>7/8$, a quoted external result whose proof is not part of the paper.

What carries the argument

The engine is a Hölder step that writes the weighted prime sum as $|S_w(\alpha)|^2=(|S_w(\alpha)|^4)^{1/q}(|S_w(\alpha)|^2)^{1-2/q}$ and multiplies it by $K(\alpha)=\sum_{n\in A}e(-2\phi(n)\alpha)$, the exponential sum over the exceptional index set $A\subseteq\{1,\dots,N\}$ of size $Z$ whose values fail Goldbach. On the major arcs, with the smooth weight $w$ equal to 1 on $[1/8,7/8]$, a quoted asymptotic gives the representation count $R_1(m)=S(m)I_w(m)+O(X^{1-c})$, where the singular series satisfies $S(m)\ge 2C_0>0$ and $I_w(m)\gg X$, so every $m=2\phi(n)$ has many weighted prime-pair representations unless $n\in A$; summing $R(2\phi(n))=0$ then forces the minor-arc integral of $S_w^2K$ to be large. Hölder's inequality bounds that integral by the minor-arc fourth moment of $S_w$ (quoted, needing $\theta<11/12$), the Parseval identity $\int|S_w|^2\ll XL$, and a $q$-th moment estimate for $K$ that is uniform in the subset $A$ and essentially sharp in its size, $\int|K|^q\ll N^{b+\varepsilon}Z^{q/2}$. The three suppliers of that estimate are Lemma G ($q=4$, $b=0$, via the factorization $(x-y)(a(x+y)+b)=t$), Theorem F ($q=8$, $b=1$, for degree at least 3), and Corollary E ($q=k(k+1)$, $b=k(k-1)/2$, from the weighted Vinogradov mean-value theorem). The outcome is the master bound $Z\ll N^{(k+2b)/q+\varepsilon}$, and the degree-wise minima of $(k+2b)/q$ are exactly $\alpha_2=1/2$, $\alpha_3=5/8$, $\alpha_4=3/4$, $\alpha_k=1-1/(k+1)$.

Extended reading notes

Core claim

The central claim, Theorem 1, is: for every fixed $\phi\in\mathbb{Z}[x]$ of degree $k\ge 2$ with positive leading coefficient, and every $\varepsilon>0$, the count $E_{k,\phi}(N)$ of indices $n\le N$ for which $2\phi(n)$ is not a sum of two primes satisfies $E_{k,\phi}(N)\ll_{\phi,\varepsilon} N^{\alpha_k+\varepsilon}$, with $\alpha_2=1/2$, $\alpha_3=5/8$, $\alpha_4=3/4$, and $\alpha_k=1-1/(k+1)$ for $k>4$. The paper is explicit that the proof is conditional: it needs $\theta<11/12$, where $\theta$ is the supremum of the real parts of nontrivial zeros of all primitive Dirichlet $L$-functions, and it imports this hypothesis from an external source in the stronger form $\theta\le 7/8$ (its Theorem A, quoted in Section 2). Given that input, two quoted theorems do the analytic work: a minor-arc fourth-moment bound for the weighted prime sum, $\int_{\mathfrak m}|S_w(\alpha)|^4\,d\alpha\ll X^{5/2}(\log X)^{100}$, and a major-arc asymptotic relating the representation integral to the singular series $S(m)$ times a weight convolution of size $\gg X$. The new contribution is the polynomial side: matching the prime fourth moment against mean-value estimates for exponential sums over arbitrary subsets of polynomial sequences, the proof shows the exceptional set for $2\phi(n)$ has size at most $N^{(k+2b)/q+\varepsilon}$, and the three families of subset moment estimates — an elementary divisor argument for degree 2, an eighth-moment discrete-restriction bound for degrees 3 and 4, and Vinogradov mean-value estimates for $k>4$ — convert that master bound into the exponents of Theorem 1.

Load-bearing premise

The load-bearing premise is Theorem A — the assertion, quoted from an external preprint and neither proved nor summarized in this paper, that all primitive Dirichlet $L$-functions are zero-free in the fixed half-plane $\operatorname{Re}s>7/8$; the paper's own summary is that the proof needs only $\theta<11/12$ and Theorem A supplies it, so if that quoted premise fails, the minor-arc fourth-moment bound and the major-arc asymptotic — and with them the whole theorem — have no support.

Editorial extensions

If this is right

For every fixed integer polynomial $\phi$ of degree $k\ge 2$, the even numbers $2\phi(n)$ are sums of two primes for all but $N^{\alpha_k+o(1)}$ of the indices $n\le N$, a set of density zero; for quadratic $\phi$ the saving is strongest, at most $N^{1/2+o(1)}$ exceptions.
The exponents are the first explicit power savings for Goldbach numbers among polynomial values at every degree, improving on the earlier $N(\log N)^{-A}$ and $N^{1-c_0/k}$ bounds, whose previous saving in the exponent was not made explicit.
For the thin sequences $2\lfloor a(n+\beta)^r\rfloor^k$, for almost every $\beta\in[0,1]$ and any $r>1/(1-\alpha_k)$ — for example $r>2$ when $k=2$ — every sufficiently large term is a sum of two primes, and the sequence's counting function is asymptotic to $(X/2a^k)^{1/(kr)}$.
Because $\alpha_k=1-1/(k+1)$ for $k>4$ approaches 1 as the degree grows, the saving over the trivial bound $N$ shrinks with $k$; the best constants occur at low degree, where the eighth-moment estimate beats the general mean-value formula.

Reading between the lines

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

If the quoted zero-free half-plane is ever proved and the minor-arc fourth-moment exponent pushed below $5/2$, the displayed chain (4.6)–(4.7) transfers that improvement linearly: replacing the $X^{5/2}$ factor by $X^{5/2-\delta}$ lowers every $\alpha_k$ by $2k\delta/q$, so the paper's exponents are best read as a scaffold attached to the prime fourth moment rather than as a hard floor.
The explicit-$\beta$ question the authors leave open admits a direct computational probe: for $\beta=0$, $k=2$, $r=3$, one can test whether every large $n$ gives $2n^6$ as a sum of two primes; a single counterexample would answer the open question in the negative, while agreement over a long range would only sharpen it.
For $k\ge 5$ the binding constraint is the sparse-subset mean value, not the prime machinery: any uniform mean-value theorem of the shape $N^{b+\varepsilon}Z^{q/2}$ would feed the same master bound, so progress on discrete restriction for sparse polynomial sequences would lower $\alpha_k$ without altering the circle-method side of the proof.

Original abstract

Let $\phi$ be a fixed polynomial with integer coefficients, positive leading coefficient and degree $k\geq2$. We prove that the values $2\phi(n)$, $1\leq n\leq N$, can be written as the sum of two primes, apart from at most $N^{\alpha_k+o(1)}$ exceptional indices, where $\alpha_k<1$. For quadratic polynomials the exponent is $1/2$, for cubic polynomials $5/8$, for quartic polynomials $3/4$, and for $k>4$ it is $1-1/(k+1)$. The proof combines a fourth-moment estimate over primes with mean-value estimates over arbitrary subsets of polynomial sequences.

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, 2026-11-09: 1 request, 3,700 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.70 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.08 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Let $\phi$ be a fixed polynomial with integer coefficients, positive leading coefficient and degree $k\geq2$. We prove that the values $2\phi(n)$, $1\leq n\leq N$, can be written as the sum of two primes, apart from at most $N^{\alpha_k+o(1)}$ exceptional indices, where $\alpha_k<1$. For quadratic polynomials the exponent is $1/2$, for cubic polynomials $5/8$, for quartic polynomials $3/4$, and for $k>4$ it is $1-1/(k+1)$. The proof combines a fourth-moment estimate over primes with mean-value estimates over arbitrary subsets of polynomial sequences.

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 $\phi$ be a fixed polynomial with integer coefficients, positive leading coefficient and degree $k\geq2$.	background 0.97	0.09	0.02	asserted 0.53
s1	We prove that the values $2\phi(n)$, $1\leq n\leq N$, can be written as the sum of two primes, apart from at most $N^{\alpha_k+o(1)}$ exceptional indices, where $\alpha_k<1$.	result 0.97	0.96	0.95	asserted 1.00
s2	For quadratic polynomials the exponent is $1/2$, for cubic polynomials $5/8$, for quartic polynomials $3/4$, and for $k>4$ it is $1-1/(k+1)$.	result 1.00	0.88	0.14	asserted 1.00
s3	The proof combines a fourth-moment estimate over primes with mean-value estimates over arbitrary subsets of polynomial sequences.	method 1.00	0.88	0.02	asserted 0.98

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

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 (10)

[1]

J. Br\"udern, K. Kawada and T. D. Wooley, Additive representation in thin sequences, II: The binary Goldbach problem, Mathematika 47 (2000), 117--125

View on Pith 2000
[2]

G. H. Hardy and J. E. Littlewood, Some problems of `Partitio Numerorum', V: A further contribution to the study of Goldbach's problem, Proc. London Math. Soc. (2) 22 (1924), 46--56

View on Pith 1924
[3]

K. Hughes and T. D. Wooley, Discrete restriction for (x,x^3) and related topics, International Mathematics Research Notices 2022 , no. 20, 15612--15631. doi:10.1093/imrn/rnab113 https://doi.org/10.1093/imrn/rnab113. Theorem 4.1 and Conjecture 4.2

View on Pith doi:10.1093/imrn/rnab113 2022
[4]

H. Li and Z. Liu, Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes, arXiv:2610.26756v1 https://arxiv.org/abs/2610.26756v1, 19 October 2026. Theorem 1, Proposition 4.1 and Section 2

View on Pith Pith review arXiv 2026
[5]

OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s>7/8 , OpenAI Math Release preprint, 30 September 2026, family 003, Theorem 1.1. Archived repository version https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf at commit fd4aeeb2ee4f

View on Pith 2026
[6]

OpenAI, The quasi-Riemann hypothesis: scope of the Lean formalization, OpenAI Math Release, family 003. Formalization scope note https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/003.md at the same repository commit

View on Pith
[7]

A. Perelli, Goldbach numbers represented by polynomials, Rev. Math. Iberoamericana 12 (1996), 477--490

View on Pith 1996
[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 Arith. 210 (2023), 53--94

View on Pith 2023
[9]

J. Pintz and I. Alth\"ofer, The Goldbach exceptional set: an exponent 7/12 and conditional refinement, arXiv:2610.15559 https://arxiv.org/abs/2610.15559

View on Pith Pith review arXiv
[10]

T. D. Wooley, Nested efficient congruencing and relatives of Vinogradov's mean value theorem, Proc. London Math. Soc. (3) 118 (2019), no. 4, 942--1016. doi:10.1112/plms.12204 https://doi.org/10.1112/plms.12204. Theorem 1.1 and Corollary 1.4. Author's version https://www.math.purdue.edu/ twooley/publ/2017nec.pdf

View on Pith doi:10.1112/plms.12204 2019

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] [1]

J. Br\"udern, K. Kawada and T. D. Wooley, Additive representation in thin sequences, II: The binary Goldbach problem, Mathematika 47 (2000), 117--125

View on Pith 2000

[2] [2]

G. H. Hardy and J. E. Littlewood, Some problems of `Partitio Numerorum', V: A further contribution to the study of Goldbach's problem, Proc. London Math. Soc. (2) 22 (1924), 46--56

View on Pith 1924

[3] [3]

K. Hughes and T. D. Wooley, Discrete restriction for (x,x^3) and related topics, International Mathematics Research Notices 2022 , no. 20, 15612--15631. doi:10.1093/imrn/rnab113 https://doi.org/10.1093/imrn/rnab113. Theorem 4.1 and Conjecture 4.2

View on Pith doi:10.1093/imrn/rnab113 2022

[4] [4]

H. Li and Z. Liu, Fourth moments of prime exponential sums and exceptional sets for sums and differences of primes, arXiv:2610.26756v1 https://arxiv.org/abs/2610.26756v1, 19 October 2026. Theorem 1, Proposition 4.1 and Section 2

View on Pith Pith review arXiv 2026

[5] [5]

OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s>7/8 , OpenAI Math Release preprint, 30 September 2026, family 003, Theorem 1.1. Archived repository version https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf at commit fd4aeeb2ee4f

View on Pith 2026

[6] [6]

OpenAI, The quasi-Riemann hypothesis: scope of the Lean formalization, OpenAI Math Release, family 003. Formalization scope note https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/003.md at the same repository commit

View on Pith

[7] [7]

A. Perelli, Goldbach numbers represented by polynomials, Rev. Math. Iberoamericana 12 (1996), 477--490

View on Pith 1996

[8] [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 Arith. 210 (2023), 53--94

View on Pith 2023

[9] [9]

J. Pintz and I. Alth\"ofer, The Goldbach exceptional set: an exponent 7/12 and conditional refinement, arXiv:2610.15559 https://arxiv.org/abs/2610.15559

View on Pith Pith review arXiv

[10] [10]

T. D. Wooley, Nested efficient congruencing and relatives of Vinogradov's mean value theorem, Proc. London Math. Soc. (3) 118 (2019), no. 4, 942--1016. doi:10.1112/plms.12204 https://doi.org/10.1112/plms.12204. Theorem 1.1 and Corollary 1.4. Author's version https://www.math.purdue.edu/ twooley/publ/2017nec.pdf

View on Pith doi:10.1112/plms.12204 2019