Primes between 8th powers?

Matt Visser (Victoria University of Wellington)

REVIEW

Primes between 8th powers?

Matt Visser (Victoria University of Wellington)

Number Theory · Submitted 14 Oct 2026 · 18 references

Pith's verdict: Useful conditional note with clean Sections 3–5, but Theorem 6 leans on a Legendre verification treated as an Oppermann verification, so one headline claim is unsupported as written.

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arXiv ID2610.20058 (version 1)

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

CitationMatt Visser (Victoria University of Wellington). "Primes between 8th powers?." arXiv:2610.20058 (2026). https://pith.science/paper/EAUTICEU

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Claim paper

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Editor's note

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The paper is a short conditional note, not a deep number theory contribution. Assuming the claimed QRH 7/8 plus the Fiori–Jaskari explicit psi bound, it derives primes in (n^8,(n+1)^8), (n^8,n^8+8n^7), (n^8,n^8+n^7), and (x,x+x^(7/8)). The first three are clean, largely substitution into a known theta inequality; the fourth has a broken low-range verification.

Credit where it is due: Visser is upfront that everything is contingent on an unproved, AI-origin claim, and he separates hypothesis from deduction cleanly. Sections 3–5 are straightforward algebra from (2.1) and the Chebyshev relation; the positivity thresholds are plausible, and I found no hidden circular step. The small finite checks, such as pi(2^8)=54, pi(3^8)=847, and the nextprime checks up to n=181, are appropriate in kind. Making the standard consequence explicit has some value while the QRH situation is evolving.

The main soft spot is Theorem 6. The analytic inequality only becomes positive for x>1.122e18, so the proof needs computational coverage of [1e8, 1.125e18]. The bridge offered is the claim that the upper half of Oppermann's conjecture, pi(x+sqrt(x))>pi(x), has been verified below 1.10e27, with Sorenson–Webster cited. That citation verifies Legendre's conjecture: for integer n, a prime between n^2 and (n+1)^2. That does not imply the real-x statement. When x sits just below (n+1)^2, the prime Legendre guarantees can lie below x and says nothing about (x, x+sqrt(x)]. So the range is unsupported as written. The fix is concrete—supply a genuine real-x Oppermann verification or shrink the claim—but it is load-bearing for one of the four headline results.

A secondary complaint: none of the finite checks is shipped as code or tables, and Section 4's "easy to check" range has no data. Those checks are probably correct, but the paper is not independently reproducible from the text. The citation pattern otherwise looks fine; [12]/[13] are the real mismatch.

Who is this for? People tracking the OpenAI QRH preprints who want the immediate interval consequences written down. It is not a new technique and not a verification of QRH. As submitted, I would desk-reject rather than spend referee time, because one headline theorem's proof depends on a misread citation. If the author supplies a correct computational bridge, the note becomes refereeable in a minor venue.

What would settle it

A reader can settle the two finite parts directly. Recompute the small ranges the paper leaves to a laptop — nextprime$(n^8) < n^8 + n^7$ for integers $16 \le n \le 181$, and $\pi(x + x^{7/8}) > \pi(x)$ on a fine grid over $1 < x \le 10^8$ — since a single violation there would refute Theorems 5 or 6 without any appeal to the quasi-Riemann hypothesis. Next, open references [12] and [13] and check whether their algorithm establishes $\pi(x + \sqrt{x}) > \pi(x)$ for all real $x < 1.10 \times 10^{27}$ or only the integer Legendre statement (a prime in $(n^2, (n+1)^2)$ for each integer $n$); if only the latter, the claimed coverage of $[10^8, (9/8)\times 10^{18}]$ that Theorem 6 needs has not been demonstrated. Independently, a numerical search of the critical strip finding a nontrivial zeta zero with real part greater than $7/8$ would falsify the hypothesis on which all four theorems depend, leaving them true-but-vacuous.

Referee report

Significance

If the external assumptions hold, the paper delivers fully explicit conditional short-interval prime theorems: the interval (n^8, n^8+n^7) has gap exponent 7/8 at height n^8, and Theorem 6 yields the same Hoheisel-type exponent uniformly for all real x > 1, which is substantially stronger than the best unconditional exponent 0.525 (Baker-Harman-Pintz) and than the unconditional statements quoted in the introduction. The strengths of the manuscript are transparency and checkability: the constants are all inherited from (2.1)-(2.2) with no fitted parameters; the thresholds in Sections 3-5 are stated explicitly to ten digits; the algebra is elementary and re-verifiable; and the conditional status of every claim is stated honestly throughout. The one load-bearing defect is the low-range bridge in Section 6, where a Legendre-conjecture verification is misdescribed as an Oppermann verification; as detailed below, this is a genuine flaw in the written proof of a headline theorem, but it is repairable by a short elementary argument from the very same cited computation, so the theorem itself is likely correct.

Major objections (1)

[Section 6, proof of Theorem 6 (paragraph after (6.6))] The proof of Theorem 6 covers the middle range x in [10^8, (9/8)*10^18] by the assertion that [x, x+x^(7/8)] contains [x, x+sqrt(x)] and that the upper half of Oppermann's conjecture, 'written as pi(x+sqrt(x)) > pi(x)', has been 'explicitly verified for x < (3.33*10^13)^2 = 1.10*10^27', citing [12], with [13] extending the range. References [12] and [13] verify Legendre's conjecture, namely the existence of a prime in (n^2, (n+1)^2) for integer n up to 3.33*10^13 and 7*10^13, respectively. That statement does not imply the real-variable inequality pi(x+sqrt(x)) > pi(x): when x lies in the lower part of a unit square interval (n^2,(n+1)^2), the Legendre prime for that interval can lie below x, and the prime delivered by the next interval can lie above x+sqrt(x); nothing in the cited computation excludes these cases. The printed argument therefore leaves all of x in [10^8, (9/8)*10^18] uncovered, and Theorem 6 is not proved as written; since Theorem 6 is one of the four headline claims, this is a load-bearing defect. The defect is, however, locally repairable within the manuscript's scope: for X >= 10^8, set n = floor(sqrt(X)); the prime q delivered by [13] in ((n+1)^2,(n+2)^2) (allowed since n <= floor(sqrt((9/8)*10^18)) is about 1.07*10^9, far below 7*10^13) satisfies X < q and q - X < (n+2)^2 - n^2 = 4n + 4 <= 4*sqrt(X) + 4 < X^(7/8), so q lies in (X, X+X^(7/8)); the last inequality is elementary for X >= 10^8. The authors should either insert this argument (or an equivalent one from a genuine consecutive-prime-gap computation) and correct the description of what [12] and [13] establish.

Minor objections (6)

[Section 5, finite check for n in [1,181]] The assertion 'Performing these checks' is not accompanied by any output. The manuscript should report, for example, the minimum of nextprime(n^8) - n^8 over 1 <= n <= 181, together with the primality routine used (the values reach roughly 1.15*10^18), so that a reader can reproduce the computation in a few minutes.
[Section 6, first bullet of the low-range check] The claim that pi(x+x^(7/8)) > pi(x) for all real x in [1,10^8] is asserted without describing the computation. The equivalent finite formulation is the verification that every consecutive-prime gap below 10^8 is smaller than x^(7/8) at the lower endpoint; the manuscript should state this formulation and report the maximal gap found.
[Section 6, statement of Theorem 6] The theorem is announced 'for positive reals x' and then restricted to (x > 1); the statement and the abstract should be aligned, with the abstract's 'reals greater than unity' being the correct formulation.
[Section 6, threshold after (6.4)] The equality 1.122210792*10^18 = (180.4094594)^8 is stated without derivation. Since (6.4) is obtained from (5.4) by the substitution y = x^(1/8), the paper should display this substitution explicitly; it both explains the threshold and clarifies the remark that Theorem 6 is 'largely a repackaging' of the result of Section 5.
[Sections 3-6, positivity thresholds] The positivity thresholds (for example 'x > 2.473174837' after (3.4), and the analogous constants after (4.4), (5.4), (6.4)) are asserted to ten significant figures without any indication of how they were obtained. The expressions are not manifestly monotone by inspection; a short appendix with a monotonicity verification or the underlying script would make the paper self-contained and would allow referees to confirm the constants.
[Section 2, reference [9]] Reference [9] is a Wikipedia page; a primary source for the psi-theta bound (2.2) would be preferable.

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The paper

The paper claims

If every nontrivial zero of the Riemann zeta function lies at real part 7/8 or below, the paper proves primes must occur between consecutive eighth powers and inside every interval (x, x + x^{7/8}) for x > 1.

The reading

The paper takes an unproven claim — the quasi-Riemann hypothesis at exponent 7/8, that every nontrivial zero $\rho$ of the Riemann zeta function satisfies $\operatorname{Re}(\rho) \le 7/8$ — and asks exactly what it would buy for the location of primes. Assuming it, the paper proves that for every positive integer $n$ there is a prime in each of the intervals $(n^8, (n+1)^8)$, $(n^8, n^8 + 8n^7)$, and $(n^8, n^8 + n^7)$, and for every real $x > 1$ a prime in $(x, x + x^{7/8})$. The point is explicitness and uniformity: unconditional results currently guarantee primes only between consecutive 86th powers for all $n$, or between cubes for enormous $n$, and a Hoheisel-type interval of length $x^{0.525}$ only for unspecified sufficiently large $x$. Here the four guarantees hold for every $n$ and every $x > 1$, with every constant written out and each small range assigned to a finite computation or a cited verification. Everything is contingent on a hypothesis the paper itself does not verify, and which it explicitly describes as claimed rather than established.

What carries the argument

The carrier of the argument is an explicit pair of two-sided bounds on the Chebyshev functions. From the assumed zero-free half-plane $\operatorname{Re}(s) > 7/8$, a companion preprint supplies $|\psi(x) - x| < 0.04934\,x^{3/4}\ln(x)^2 + 0.02636425\,x^{7/8}$ for $x \ge e^3$; the paper combines this with the elementary estimate $0 < \psi(x) - \vartheta(x) < \sqrt{x}\,\ln(x)^2/(2\ln 2)$ to obtain asymmetric upper and lower bounds on $\vartheta(x)$ (equations 2.4 and 2.5). The mechanism that turns these bounds into primes is monotonicity: whenever the derived lower estimate for $\vartheta(b) - \vartheta(a)$ is positive, at least one prime lies in $(a,b]$, because $\vartheta$ jumps by $\ln p$ exactly at primes. Substituting the four endpoint pairs $((n+1)^8, n^8)$, $(n^8 + 8n^7, n^8)$, $(n^8 + n^7, n^8)$, and $(x + x^{7/8}, x)$, the error terms scale with the interval length at exactly the 7/8 rate, and the constants are small enough that the difference is provably positive for $x \ge 3$, $x \ge 5$, $x \ge 181$, and $x > 1.1222\times 10^{18}$, respectively, after which each remaining finite range is cleared by explicit computation.

Extended reading notes

Core claim

On the paper's own terms, the discovery is a chain of four conditional theorems. Assuming QRH 7/8 — that all nontrivial zeros $\rho$ of the Riemann zeta function satisfy $\operatorname{Re}(\rho) \le 7/8$ — the paper establishes, for all $n \in \mathbb{Z}^+$: $\pi((n+1)^8) > \pi(n^8)$, $\pi(n^8 + 8n^7) > \pi(n^8)$, and $\pi(n^8 + n^7) > \pi(n^8)$, and for all real $x > 1$: $\pi(x + x^{7/8}) > \pi(x)$. Each inequality states that at least one prime exists inside an explicit window; the narrowest integer window has relative length $1/n$, and the real window has relative length $x^{-1/8}$, so all four windows shrink to a vanishing fraction of the location as the numbers grow. The derivation is completely explicit: the hypothesis enters only through a companion preprint's two-sided estimate on the second Chebyshev function $\psi(x)$, which the paper converts into asymmetric bounds on the first Chebyshev function $\vartheta(x)$ and evaluates at the endpoints of each window, with the finite leftover ranges disposed of by direct computation. The paper states plainly that none of this stands without the hypothesis, which it describes as claimed by an unreleased AI-generated preprint collection rather than proved.

Load-bearing premise

Everything rests on the unproven claim called QRH 7/8 — that every nontrivial zero of the Riemann zeta function has real part at most 7/8 — and, within the derivation, Theorem 6 also leans on the assertion that reference [12] verifies the real-variable inequality $\pi(x + \sqrt{x}) > \pi(x)$ for $x$ up to $1.10 \times 10^{27}$, whereas the cited computation in fact verifies Legendre's conjecture for integer $n$, a weaker statement that does not imply the real-variable form the proof requires.

Editorial extensions

If this is right

If QRH 7/8 is genuine, every gap $(n^8, (n+1)^8)$ provably contains a prime, an eighth-power analogue of Legendre's conjecture, whereas the current unconditional guarantee only reaches consecutive 86th powers.
Primes are also guaranteed within $n^7$ of $n^8$, a relative window of length $1/n$; for large $n$ this is far shorter than any interval certified unconditionally at such small starting values.
The real-variable statement $\pi(x + x^{7/8}) > \pi(x)$ for every $x > 1$ would make short-interval prime guarantees uniform and explicit, replacing 'for sufficiently large $x$' with a bound that holds everywhere.
As a direct corollary, prime gaps would satisfy $p_{k+1} - p_k \le p_k^{7/8}$ for every prime $p_k > 1$; the exponent is weaker than the best asymptotic unconditional results, but the bound would hold for all primes with explicit constants.

Reading between the lines

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

The paper leaves implicit that its template is a general conversion: any claimed zero-free half-plane with exponent $\theta$ feeds the same endpoint-difference argument and yields primes in $(x, x + x^\theta)$ for all $x > 1$, so the weaker claimed exponents $11/12$ and $34999/40000$ would give the same theorems with longer windows and shorter computation ranges.
Not in the paper, but worth stating: the small-number checks are unconditional and could be reproduced on a laptop — nextprime$(n^8) < n^8 + n^7$ for $16 \le n \le 181$, and $\pi(x + x^{7/8}) > \pi(x)$ on a grid over $1 < x \le 10^8$ — and a single failure there would refute the corresponding theorem regardless of the zeta function.
Read as an algorithm rather than as one result, the paper shows that the exponent of the zero-free half-plane and the exponent of the guaranteed prime window coincide (both 7/8 here); eighth powers are one concrete output, and the really transferable content is the explicit procedure from zero location to prime location.

Original abstract

OpenAI (using a non-public "internal frontier model'') has recently (6 October 2026) made a number of remarkable claims, including some claims regarding the quasi-Riemann hypothesis (QRH). While there are good reasons to be cautious regarding these claims, one can nevertheless ask what (if true) their implications would be. While there is widespread agreement that any version of the QRH will provide various improved bounds on the location of the prime numbers, I feel that there is some value in making these improvements explicit. Specifically, if we assume validity of the claimed 7/8 version of the QRH then I shall demonstrate (for all $n\in \mathbb{Z^+}$): 1) the existence of primes between eighth powers, in the interval $(n^8,(n+1)^8)$; 2) the existence of primes in the shorter interval $(n^8,n^8+8n^7)$; 3) the existence of primes in the even shorter interval $(n^8,n^8+n^7).$ Similarly for reals greater than unity $x>1$ one can demonstrate: 4) the existence of primes in the Hoheisel-style interval $(x,x+x^{7/8})$. All of this is contingent on the validity of the claimed 7/8 version of the QRH.

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, 2026-10-15: 1 request, 4,542 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.96 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.84 · compares with prior results 0.12 · 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.

OpenAI (using a non-public "internal frontier model'') has recently (6 October 2026) made a number of remarkable claims, including some claims regarding the quasi-Riemann hypothesis (QRH). While there are good reasons to be cautious regarding these claims, one can nevertheless ask what (if true) their implications would be. While there is widespread agreement that any version of the QRH will provide various improved bounds on the location of the prime numbers, I feel that there is some value in making these improvements explicit. Specifically, if we assume validity of the claimed 7/8 version of the QRH then I shall demonstrate (for all $n\in \mathbb{Z^+}$): 1) the existence of primes between eighth powers, in the interval $(n^8,(n+1)^8)$; 2) the existence of primes in the shorter interval $(n^8,n^8+8n^7)$; 3) the existence of primes in the even shorter interval $(n^8,n^8+n^7).$ Similarly for reals greater than unity $x>1$ one can demonstrate: 4) the existence of primes in the Hoheisel-style interval $(x,x+x^{7/8})$. All of this is contingent on the validity of the claimed 7/8 version of the QRH.

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	OpenAI (using a non-public "internal frontier model'') has recently (6 October 2026) made a number of remarkable claims, including some claims regarding the quasi-Riemann hypothesis (QRH).	background 1.00	0.04	0.02	asserted 1.00
s1	While there are good reasons to be cautious regarding these claims, one can nevertheless ask what (if true) their implications would be.	background 0.76	0.06	0.02	hedged possibility 0.62
s2	While there is widespread agreement that any version of the QRH will provide various improved bounds on the location of the prime numbers, I feel that there is some value in making these improvements explicit.	background 0.63	0.09	0.04	asserted 0.58
s3	Specifically, if we assume validity of the claimed 7/8 version of the QRH then I shall demonstrate (for all $n\in \mathbb{Z^+}$): 1) the existence of primes between eighth powers, in the interval $(n^8,(n+1)^8)$; 2) the existence of primes in the shorter interval $(n^8,n^8+8n^7)$; 3) the existence of primes in the even shorter interval $(n^8,n^8+n^7).$ Similarly for reals greater than unity $x>1$ one can demonstrate: 4) the existence of primes in the Hoheisel-style interval $(x,x+x^{7/8})$.	result 0.70	0.82	0.91	asserted 0.89
s4	All of this is contingent on the validity of the claimed 7/8 version of the QRH.	limitation 1.00	0.12	0.06	asserted 0.63

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 0.93.

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 (18)

[12]

Jonathan Sorenson, Jonathan Webster, “An algorithm and computation to verify Legendre’s Conjecture up to 3.33×10 13”, arXiv: 2401.13753 [math.NT] (24 Jan 2024)

View on Pith Pith review arXiv 2024
[13]

Jonathan Sorenson, Jonathan Webster, “An algorithm and computation to verify Legendre’s Conjecture up to 7×10 13”, Research in Number Theory11(2025) 4, https://doi.org/10.1007/s40993-024-00589-4 – 9 –

View on Pith doi:10.1007/s40993-024-00589-4 2025
[8]

Andrew Fiori and Mikko Jaskari, “Explicit bounds for the prime number theorem”, arXiv: 2609.23222 [math.NT], (8 October 2026)

View on Pith Pith review arXiv 2026
[1]

OpenAI, “Sharing AI progress in mathematics”, https://openai.com/index/sharing-ai-progress-in-mathematics/ Specifically, see:https://github.com/openai/math/tree/main/preprints, (6 October 2026). [2]https://github.com/openai/math/blob/main/history.md (Three articles withdrawn, another twelve corrected, as of 14 October 2026.)

View on Pith 2026
[3]

Alexander Bastounis, Fabian Circelli, Anders C. Hansen, “Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs”, arXiv: 2610.08144 [math.AP], (6 October 2026)

View on Pith Pith review arXiv
[4]

OpenAI, “The quasi-Riemann hypothesis”, https://github.com/openai/math/blob/main/preprints/ The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf, (5 October 2026)

View on Pith 2026
[5]

OpenAI, “The quasi-Riemann hypothesis: A Zero-Free Half-Plane Re(s)>7/8”, https://github.com/openai/math/blob/main/preprints/ The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf, (30 September 2026)

View on Pith 2026
[6]

Shiva Kintali, “A Short Proof of the Quasi-Riemann Hypothesis”, https://shivakintali.github.io/papers/QRH.pdf, (7 October 2026)

View on Pith 2026
[7]

Baiying Liu, “Slightly improved zero-free half-planes for the quasi-Riemann hypothesis”, arXiv: 2610.12234 [math.NT], (8 October 2026)

View on Pith Pith review arXiv 2026
[9]

See for examplehttps://en.wikipedia.org/wiki/Chebyshev_function

View on Pith
[10]

Guido Hoheisel, “Primzahlprobleme in der Analysis”, Sitzungsberichte Berliner Akad. d. Wiss., (1930) 580-588

View on Pith 1930
[11]

Matt Visser, “Effective short intervals containing primes”, arXiv: 2508.18786 math.NT

View on Pith Pith review arXiv
[14]

Ethan Simpson Lee, “Minimal zero-free regions for results on primes between consecutive perfect k-th powers”, arXiv: 2602.14340 [math.NT] (3 March 2026). Acta Arith. 2026 (in press)

View on Pith arXiv 2026
[15]

Michaela Cully-Hugill, “Primes between consecutive powers”, Journal of Number Theory,247(2023) 100–117. doi:10.1016/j.jnt.2022.12.002 [arXiv: 2107.14468 [math.NT]]

View on Pith Pith review arXiv 2023
[16]

Michaela Cully-Hugill and Daniel R. Johnston, “On the error term in the explicit formula of Riemann–von Mangoldt”, International Journal of Number Theory19 # 6(2023) 1205–1228 doi:10.1142/S1793042123500598 [arXiv: 2111.10001 [math.NT]]

View on Pith Pith review arXiv 2023
[17]

Michaela Cully-Hugill and Daniel R. Johnston, “On the error term in the explicit formula of Riemann–von Mangoldt. II.”, Funct. Approx. Comment. Math.73 # 2(2025) 223–242 . doi:10.7169/facm/241110-18-11 [arXiv: 2402.04272 [math.NT]]

View on Pith arXiv 2025
[18]

Adrian Dudek, “An Explicit Result for Primes Between Cubes”, Funct. Approx. Comment. Math.55 # 2(2016) 177–197. doi:10.7169/facm/2016.55.2.3 [arXiv: 1401.4233 [math.NT]]

View on Pith Pith review arXiv 2016
[19]

R. C. Baker, G. Harman, and J. Pintz, “The Difference Between Consecutive Primes, II”, Proc. London Math. Soc.83 # 3(2001) 532–562. – 10 –

View on Pith 2001

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] [12]

Jonathan Sorenson, Jonathan Webster, “An algorithm and computation to verify Legendre’s Conjecture up to 3.33×10 13”, arXiv: 2401.13753 [math.NT] (24 Jan 2024)

View on Pith Pith review arXiv 2024

[2] [13]

Jonathan Sorenson, Jonathan Webster, “An algorithm and computation to verify Legendre’s Conjecture up to 7×10 13”, Research in Number Theory11(2025) 4, https://doi.org/10.1007/s40993-024-00589-4 – 9 –

View on Pith doi:10.1007/s40993-024-00589-4 2025

[3] [8]

Andrew Fiori and Mikko Jaskari, “Explicit bounds for the prime number theorem”, arXiv: 2609.23222 [math.NT], (8 October 2026)

View on Pith Pith review arXiv 2026

[4] [1]

OpenAI, “Sharing AI progress in mathematics”, https://openai.com/index/sharing-ai-progress-in-mathematics/ Specifically, see:https://github.com/openai/math/tree/main/preprints, (6 October 2026). [2]https://github.com/openai/math/blob/main/history.md (Three articles withdrawn, another twelve corrected, as of 14 October 2026.)

View on Pith 2026

[5] [3]

Alexander Bastounis, Fabian Circelli, Anders C. Hansen, “Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs”, arXiv: 2610.08144 [math.AP], (6 October 2026)

View on Pith Pith review arXiv

[6] [4]

OpenAI, “The quasi-Riemann hypothesis”, https://github.com/openai/math/blob/main/preprints/ The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf, (5 October 2026)

View on Pith 2026

[7] [5]

OpenAI, “The quasi-Riemann hypothesis: A Zero-Free Half-Plane Re(s)>7/8”, https://github.com/openai/math/blob/main/preprints/ The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf, (30 September 2026)

View on Pith 2026

[8] [6]

Shiva Kintali, “A Short Proof of the Quasi-Riemann Hypothesis”, https://shivakintali.github.io/papers/QRH.pdf, (7 October 2026)

View on Pith 2026

[9] [7]

Baiying Liu, “Slightly improved zero-free half-planes for the quasi-Riemann hypothesis”, arXiv: 2610.12234 [math.NT], (8 October 2026)

View on Pith Pith review arXiv 2026

[10] [9]

See for examplehttps://en.wikipedia.org/wiki/Chebyshev_function

View on Pith

[11] [10]

Guido Hoheisel, “Primzahlprobleme in der Analysis”, Sitzungsberichte Berliner Akad. d. Wiss., (1930) 580-588

View on Pith 1930

[12] [11]

Matt Visser, “Effective short intervals containing primes”, arXiv: 2508.18786 math.NT

View on Pith Pith review arXiv

[13] [14]

Ethan Simpson Lee, “Minimal zero-free regions for results on primes between consecutive perfect k-th powers”, arXiv: 2602.14340 [math.NT] (3 March 2026). Acta Arith. 2026 (in press)

View on Pith arXiv 2026

[14] [15]

Michaela Cully-Hugill, “Primes between consecutive powers”, Journal of Number Theory,247(2023) 100–117. doi:10.1016/j.jnt.2022.12.002 [arXiv: 2107.14468 [math.NT]]

View on Pith Pith review arXiv 2023

[15] [16]

Michaela Cully-Hugill and Daniel R. Johnston, “On the error term in the explicit formula of Riemann–von Mangoldt”, International Journal of Number Theory19 # 6(2023) 1205–1228 doi:10.1142/S1793042123500598 [arXiv: 2111.10001 [math.NT]]

View on Pith Pith review arXiv 2023

[16] [17]

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