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REVIEW 4 major objections 5 minor 31 references

New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that a systematic database of exponent relations yields four new exponent pairs, new zero-density bounds, and new additive-energy bounds for the Riemann zeta-function.

desk verdict A serious results paper with genuinely new numerical bounds, but the headline claims lean on an uncertified database computation and need a versioned, checkable artifact before they should be accepted. read the letter →

arxiv 2501.16779 v1 pith:SUWDLDM5 submitted 2025-01-28 math.NT

Terence Tao , Tim Trudgian , Andrew Yang This is my paper · ORCID
classification math.NT MSC 11L0711M0611T23
keywords exponentpairszerodensityestimatesadditiveenergyRiemannzeta-functionlargevaluetheoremscomputer-assistedproofexponentialsumsanalyticnumbertheory
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 argues that the best-known bounds on exponential-sum, zero-density, and additive-energy exponents for the Riemann zeta-function can be improved by systematically collecting every known relation among these exponents and running a computer search over their consequences. It reports four new exponent pairs, several new zero density estimates, and new upper bounds on the additive energy of zeroes of the zeta-function. The point is that these are not new analytic inputs but optimal combinations of existing theorems, made visible by a database that abstracts away the routine optimization. If the claims are right, then any future improvement to a single upstream bound can be propagated automatically to the best implied bounds across the whole network.

What carries the argument

The central object is the Analytic Number Theory Exponent Database (ANTEDB), a database in which each known theorem or conjecture about an exponent is stored both as human-readable text and as an executable object with dependencies that specify how the result is proved. The carrying mechanism is polytope optimisation: each relation among exponents (an exponent pair, a large value estimate, a zero density implication, a subdivision or power-raising rule) is converted into a convex region in the space of possible exponent tuples, and intersecting these regions yields the best bounds implied by the stored inputs. A 'cheap non-standard analysis' asymptotic formalism keeps epsilon losses uniform throughout. The output is a set of implied bounds, which the routine can convert back into machine-checkable derivations and, in many cases, human-readable proofs.

What would settle it

An independent re-implementation that starts from the same stated inequalities and recomputes the polytope intersections would settle the central claim: if any of the four exponent pairs in Theorem 20 or any bold zero-density entry in Table 2 is not implied by the stated inputs, the paper's conclusion fails.

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Extended reading notes

Core claim

In the paper's own terms, the central discovery is that four points in the exponent-pair triangle—$(89/1282,997/1282)$, $(652397/9713986,7599781/9713986)$, $(10769/351096,609317/702192)$, and $(89/3478,15327/17390)$—are genuine exponent pairs, that the zero density bounds marked in bold in Table 2 are consequences of the stated large value inputs, and that for $7/10 \le \sigma <1$ the additive energy exponent $A^*(\sigma)$ obeys the piecewise bounds recorded in Theorem 64. These results are reached by encoding the known bounds and the relations between them in a database, representing each relation as a polytope of feasible exponent tuples, and intersecting these polytopes to find the best implied bounds. The paper presents human-readable proofs for the new claims where possible, and stores machine-readable derivations for the rest.

Load-bearing premise

The load-bearing premise is that the computer code implementing the database correctly encodes every cited theorem's hypotheses and performs the optimisation without error; a bug in that code could invalidate the new bounds even though each individual input theorem is true.

Editorial extensions

If this is right

  • The four new exponent pairs immediately improve every bound that depends on exponent pairs via the standard A/B/C/D processes, including the $\beta(\alpha)$ table and the large value estimates derived from it.
  • The new zero density estimates sharpen the best known upper bounds on $N(\sigma,T)$ in several ranges, bringing known bounds closer to the density hypothesis without reaching it.
  • The additive energy bounds in Theorem 64 improve the classical estimates for $A^*(\sigma)$ over the stated intervals, which is relevant to the distribution of primes in short intervals.
  • Within the database, future improvements in any single exponent input automatically yield optimised improvements to all dependent exponents, making the routine propagation of new bounds largely mechanical.

Reading between the lines

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

  • Beyond the paper: the same machinery could be pointed at other exponent collections, such as those for $L$-functions or for prime gaps, and the paper notes this as a possible expansion; one testable extension is to run the optimiser on the announced $L$-function analogues.
  • Beyond the paper: the heuristic in Section 6.2 suggests that the bottleneck for zero density bounds is the value of $LV(\sigma,\tau_0)$; checking the database's polytope against the Montgomery conjecture threshold would show how close current methods come to the heuristic limit.
  • Beyond the paper: because the uncertified code is the weakest link, an independent implementation or formal verification of the optimisation would be a direct way to convert these numerical claims into fully certified theorem status.
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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

4 major / 5 minor

Summary. The paper introduces the Analytic Number Theory Exponent Database (ANTEDB), a Python-based system for recording known bounds and relations among exponents in analytic number theory and for optimizing them via polytope operations. Using this framework, the authors claim four new exponent pairs (Theorem 20), several new zero-density estimates for the Riemann zeta-function (Table 2, via Theorems 50, 51, and 53), and new additive-energy estimates for the zeroes (Theorem 64). The paper also abstracts a number of standard relationships, such as Huxley subdivision, the duality between exponent pairs and the β function, large-value-to-zero-density conversions, and Heath-Brown's additive-energy relation, presenting many of these as human-readable lemmas.

Significance. If the new numerical claims are correct, they constitute genuine quantitative advances in three related areas: exponent pairs, zero-density estimates, and additive energy of zeta zeroes. A further strength of the paper is its systematic abstraction of relationships that often appear only implicitly in the literature, and the public availability of the ANTEDB code and database; this has the potential to make future incremental improvements easier to propagate. The paper is also commendably explicit in several places about the limits of the computer assistance (e.g., Section 1.2 concedes that ANTEDB routines are not formally certified), and many of the structural lemmas are given with human-readable proofs. However, because several headline results are delegated to computations whose correctness is not established within the manuscript, the central claims are not currently self-contained.

major comments (4)
  1. [Theorem 20; Section 1.2] The proof of Theorem 20 consists of the sentence 'The claim follows from Lemma 15 after some further computer calculation.' Since these four exponent pairs are a headline result, and since Section 1.2 explicitly states that the ANTEDB routines 'are not formally certified to be error-free', the central claim is not verified within the paper. A single miscoded hypothesis or arithmetic bug in the polytope intersection would invalidate all four pairs. Please provide either an explicit, independently checkable transcript of the computation (for instance, the exact polytope data and the piecewise-linear β bounds used for each of the four pairs) or human-readable derivations that the reader can verify without executing external code.
  2. [Lemma 14; Table 1] Lemma 14 gives only an upper bound for β(α) in terms of an exponent pair (k, ℓ), namely β(α) ≤ max{k₁ + α(ℓ₁ − k₁), 1/12 + 2α/3}. The following sentence asserts that in practice this implies D(k, ℓ) is an exponent pair, but this implication is not immediate: it requires checking, via Lemma 15, that k₁ + (ℓ₁ − k₁)α is bounded by the displayed maximum for all α ∈ [0,1], with appropriate splitting of the range of α. That verification is omitted, and Table 1 relies on the D-process in several rows. Since Theorem 20 depends on Table 1, this gap propagates to the headline exponent-pair results.
  3. [Theorem 53; Table 2] The proof of Theorem 53 asserts that S(σ) is a convex polygon ('one may verify'), lists eight exponent pairs 'found with the aid of computer assistance', and then states that applying Theorem 52 and taking a minimum yields the piecewise formulas. The manuscript does not show that each listed pair lies in the required region S(σ) on each stated σ-interval, nor does it display the endpoint computations that produce the interval boundaries (such as 2841/3016, 859/908, 1625/1692, etc.). Because Theorem 53 supplies several bold rows of Table 2, which are advertised as new zero-density estimates, this leaves a load-bearing part of the paper unverifiable without the external database. Please supply the verified optimization data or a detailed enumeration of the interval checks.
  4. [Theorem 64] Only part (i) of Theorem 64 is proved in the paper; parts (ii)–(ix) are deferred to the ANTEDB. Section 1.2 concedes that the ANTEDB routines are not formally certified to be error-free. Since the additive energy estimates are one of the three headline contributions, the current manuscript does not provide a complete proof of eight of the nine claimed cases. Even part (i) contains many 'one can check' inequalities, which are likely routine but are not shown. Please either provide complete proofs for all parts of Theorem 64 or make available a formally verified or independently checkable computation transcript that certifies the deferred parts.
minor comments (5)
  1. [Definition 11] In the definition of an exponent pair, the phrase 'for all T ≥ N ≤ 1' should presumably read 'for all T ≥ N ≥ 1'.
  2. [Definition 58] In the definition of the large value energy region, the condition 'V = N^{σ+o(1)}' appears twice; the second occurrence is redundant and should be removed or replaced.
  3. [Table 2] Several rows in Table 2 are attributed to 'Corollary 53', but the text labels the relevant result as Theorem 53. Please correct the cross-reference.
  4. [Theorem 51] In the proof of Theorem 51, the second case is stated as '16/21 ≤ σ ≤ 38/49', while the theorem statement restricts to '17/22 ≤ σ ≤ 4/5'. Please clarify whether the proof is intended for the wider range or align the statement and proof.
  5. [Various proofs, e.g., Theorem 51 and Theorem 64] The phrase 'one can check' appears in several places where numerical inequalities are asserted. For a computer-assisted paper, it would considerably aid verification if a supplementary file enumerated these inequalities or if the code scripts that certify them were clearly designated and archived.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation found: the new bounds follow from prior published theorem inputs via explicit lemmas and an optimization code; self-citations are present but not load-bearing in a circular sense.

full rationale

The paper's derivation chain is not circular. The new exponent pairs in Theorem 20 are obtained by combining known exponent pairs and beta bounds from prior literature (Bourgain, Huxley, Heath-Brown, Watt, and the authors' earlier paper [TY23]), applying the A/B/C processes, Lemma 14, and the duality Lemma 15, then intersecting feasible polytopes in the ANTEDB code. No target exponent pair is assumed as a hypothesis; the conversion from a beta bound back to an exponent pair is proved in Lemma 15 within the paper. The zero density results in Theorems 50, 51, and 53 are derived from large value theorems of Huxley, Heath-Brown, Jutila, Bourgain, and Guth-Maynard, together with Corollaries 41-44, all of which are proved in the text. The vertex choices in Theorem 51 and Theorem 53 are found by computer search, but they are explicit and followed by stated inequalities that 'one can check'; this is a computational verification step, not a circular reduction. The additive energy result Theorem 64(i) has a full human-readable proof; parts (ii)-(ix) are deferred to the ANTEDB database, which is a completeness and auditability concern, not a reduction of the conclusion to its own input. Section 1.2 explicitly concedes that the ANTEDB routines 'are not formally certified to be error-free'; this is a correctness risk, not circularity. The self-citations, [TY23] by the second and third authors and [CT21] by the first author, are prior independent results with their own proofs and are used as inputs, not as justifications that presuppose the present conclusions. No equation in the paper fits a parameter to the target result, renames a known result as new, or imports a uniqueness claim from the authors' own prior work. The score of 2 reflects only the presence of minor self-citations and the heavy reliance on an uncertified computer database; no circular step was identified.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on a network of theorems from the literature encoded in ANTEDB. The paper does not prove these inputs anew, and some computer-assisted derivations are not formally verified. No free parameters are fitted to data; the optimization variables are proof choices, not empirical fits.

assumptions (6)
  • domain assumption Exponent pair processes A, B, C preserve exponent pairs (Lemma 13).
    The new exponent pairs in Theorem 20 are generated by repeatedly applying these known processes to previously known pairs.
  • domain assumption Sargos D-process converts exponent pairs to beta bounds (Lemma 14).
    Used with Lemma 15 to generate beta estimates in Table 1 and hence the new exponent pairs.
  • domain assumption Known large value theorems: Huxley (13), Heath-Brown (14), Jutila (15), Guth-Maynard (16), and Bourgain Theorem 32.
    These theorems from the literature are inputs to the zero density optimization; the paper does not reprove them.
  • domain assumption Heath-Brown relation between additive energy and large value patterns (Theorem 63).
    This is the main external input for the new additive energy estimates in Theorem 64.
  • domain assumption Matomaki-Teravainen partial converse from zero density to large values (Lemma 40).
    Used in the heuristic converse and recorded as a relation in the ANTEDB.
  • standard math Approximate functional equation and Riemann-von Mangoldt formula.
    Used in Lemma 34(iii) and in the zero detecting arguments in Lemmas 39 and 60.

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Pith. "Pith review of New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach." pith.science (2026). https://pith.science/paper/SUWDLDM5

@misc{pith2026250116779,
  author       = {Pith},
  title        = {Pith review of: New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SUWDLDM5}},
  note         = {Machine review of arXiv:2501.16779}
}
read the original abstract

We obtain several new bounds on exponents of interest in analytic number theory, including four new exponent pairs, new zero density estimates for the Riemann zeta-function, and new estimates for the additive energy of zeroes of the Riemann zeta-function. These results were obtained by creating the Analytic Number Theory Exponent Database (ANTEDB) to collect results and relationships between these exponents, and then systematically optimising these relationships to obtain the new bounds. We welcome further contributions to the database, which aims to allow easy conversion of new bounds on these exponents into optimised bounds on other related exponents of interest.

Figures

Figures reproduced from arXiv: 2501.16779 by the authors.

Figure 1
Figure 1. Indicative plot of the new exponent pairs ( 89 1282 , 997 1282 ) and ( 652397 9713986 , 7599781 9713986 ) compared to previously known set of expo￾nent pairs (shaded grey). describe a polytope-based computation that may be easily generalised for optimis￾ing many other exponents recorded in the database. First, the starting Hypothesis set is repeatedly expanded using codified versions of Lemma 13, Lemma 14 and Lemma … view at source ↗
Figure 2
Figure 2. Indicative plot of the new exponent pairs [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. The bounds in [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Plot of bounds on A∗ (σ) in Theorem 64, against exist￾ing literature bounds. max  27 − 31σ 4 + τ ′ 2 , 71 − 79σ 8 + τ ′ 4 , 7 − 43 5 σ + 4 5 τ ′  ≤ 7(1 − σ) 3σ − 1 τ ′ for τ ′ ≥ 3σ − 1, and that max  8 − 10σ + τ ′ , 23 − 34σ + 5τ ′ 2 , 2 5 (22 − 32σ + 5τ ′ )  ≤ 7(1…

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Works this paper leans on

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