Nonuniform Distributions of Residues of Prime Sequences in Prime Moduli

David Wu

REVIEW 6 major objections 4 minor 10 references

Nonuniform Distributions of Residues of Prime Sequences in Prime Moduli

T0 review · 6 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Under one plausible average-order assumption, this paper proves that the high-order error terms in the conjectured formula for consecutive-prime residue patterns decay like $(\log_2 y)^n/(\log y)^{n/2-1}$, making the formula truncatable…

desk verdict The n≥3 truncation and the data to 10^18 are real, but the advertised rigorous bridge from Hardy-Littlewood to Lemke Oliver–Soundararajan is not delivered: the key estimate is an unproved analogue of (3.5) and the large-gap lemma is only sketched. read the letter →

arxiv 1908.07095 v1 pith:I6M7WTUS submitted 2019-08-19 math.NT

David Wu This is my paper · ORCID

classification math.NT MSC 11N0511N13

keywords primek-tuplesHardy-Littlewoodconjectureconsecutiveprimesresidueclassespatternssingularserieslargegapsanalyticnumbertheory

topics The Twin Prime Conjecture

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

Prime numbers modulo a fixed modulus are not uniformly spread among consecutive patterns: some residue pairs appear far more often than others. This paper takes a conjectured asymptotic formula for the frequency of such patterns and analyzes its central sum $D(a,b;y)$, separating it into terms by how many auxiliary forbidden positions are included. Under one plausible assumption about the average size of the modified singular series, it proves that the high-order terms decay like $O_n\big((\log_2 y)^n/(\log y)^{n/2-1}\big)$, so the formula can be truncated with a controlled error. The result is a first rigorous connection from the prime k-tuple conjecture to the observed pattern biases, and the paper extends numerical data to $x=10^{18}$ to support the refined formula.

What carries the argument

The load-bearing objects are the singular series $S_q(H)$, modified to omit primes dividing the modulus $q$, and its inclusion-exclusion variant $S_{q,0}(H)$. Equation (3.5), a known estimate for the average order of sums of $S_0(T)$ over $\ell$-subsets of $[1,h]$, is assumed to persist for $S_{q,0}$; that estimate supplies the main size of the alternating sums $A_{h,\ell}$, $B_{h,\ell}$, $C_{h,\ell}$, and $D_{h,\ell}$. The proof splits the outer sum over gaps $h$ at $h=M\log y$, using a large-gap probability bound to discard longer gaps, then evaluates the remaining sums with convexity bounds, truncated Taylor series, and finite-difference identities that telescope the four subset sums into differences of $A_{h,\ell}$. These identities translate extra factors of $\log y$ into extra decay, giving the theorem.

What would settle it

Compute the sums $\sum_{T\subset[1,h], |T|=\ell} S_{q,0}(T)$ for a fixed odd prime modulus such as $q=3$ over a range of $h$ and $\ell$, and compare them with the predicted main term $\frac{\mu_\ell}{\ell!}(-h\log h+Ah)^{\ell/2}$. If the discrepancy exceeds the allowed $O(h^{\ell/2-1/7\ell+\epsilon})$ for any $q$, the main estimate in Theorem 3.1 fails. Alternatively, check the large-gap lemma directly by testing whether the frequency of prime gaps above $c\log_2 p_N\log p_n$ is bounded by $(\log N)^{-c}$ for fixed $c$.

Watch

Extended reading notes

Core claim

The paper's central result, Theorem 3.1, is a conditional decay bound. Assuming the known average-order estimate (3.5) for sums of singular series remains valid in similar form for the modified series $S_{q,0}$, the four pieces $S_\emptyset$, $S_{\{0\}}\log y$, $S_{\{h\}}\log y$, and $S_{\{0,h\}}(\log y)^2$ are all $O_n\big((\log_2 y)^n/(\log y)^{n/2-1}\big)$, where $\log_2 y$ is the second iterated logarithm. Consequently the terms $D_n(a,b;y)$ and their tail $D_{\ge n}(a,b;y)$, which make up the sum $D(a,b;y)$ in the conjectured asymptotic for $\pi(x;q,(a,b))$, satisfy the same bound. This means the infinite sum over auxiliary prime positions can be cut off at a finite $n$ while keeping the error under control, turning the heuristic restriction to small sets into a provable approximation.

Load-bearing premise

The proof assumes that a known average-order estimate for alternating sums of the singular series, equation (3.5), continues to hold with minor corrections when the singular series is replaced by the modified series $S_{q,0}$ that excludes primes dividing the modulus $q$; this analog is not proved in the paper, and the large-gap lemma used to discard long tails is only sketched.

Editorial extensions

If this is right

The infinite sum $D(a,b;y)$ can be truncated at any fixed $n$: the tail $D_{\ge n}(a,b;y)$ has size $O_n((\log_2 y)^n/(\log y)^{n/2-1})$, so the conjectured integral for $\pi(x;q,(a,b))$ has a controlled error once the assumption is accepted.
Because the $|T|\ge 6$ pieces are negligible, the model only needs singular-series values for sets of size at most five; the paper prepares these up to $\max H\le 150$ for numerical integration.
Large gaps $h>c\log_2 y\log y$ contribute at most $(\log y)^{-c}$, so the long-range part of the pattern count cannot be the source of the observed bias.
Theorem 3.1 converts the heuristic statement that only zero- and two-element sets matter into a conditional theorem, the first rigorous bridge from the prime k-tuple conjecture to the pattern-frequency asymptotic.

Reading between the lines

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

The paper leaves implicit that if the assumed average-order estimate for $S_{q,0}$ were proved unconditionally, the entire conjectured asymptotic for $\pi(x;q,(a,b))$ would follow from the prime k-tuple conjecture, making the known biases in consecutive-prime residues a theorem rather than a heuristic.
The same truncation logic suggests the biases are a local phenomenon, controlled only by gaps up to $c\log_2 y\log y$; a numerical test would be to recompute pattern frequencies with the tail removed and check that the remaining signal matches the conjectured formula.
The method is stated for odd prime $q$, but the author says it readily generalizes to composite moduli; testing composite $q$ numerically would indicate whether the same decay rate holds there.
The curve-fitting evidence for a $O((\log_2 x)^2/(\log x)^2)$ lower-order term is plausible as the next correction in the expansion; a sharper test would compare the full truncated model at $x=10^{18}$ against data at several larger $x$.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper has a genuinely new component—the n≥3 truncation in Theorem 3.1, the data extension to 10^18, and the explicit fitted lower-order term—but the central advertised result is not a proof. Theorem 3.1 is conditional on an unproved analogue of (3.5), and the proof leans on Lemma 4.3, which the author himself says is only sketched and “not fully rigorous.” So the abstract’s phrase “begin to rigorously connect” overstates what is established.

What is genuinely good: the decomposition of D into S_∅, S_{0}, S_{h}, and S_{0,h} is a sensible way to organize the heuristic, and Lemma 4.4 is a clean combinatorial identity with an exact proof. The numerical extension to 10^18 is useful, and the author is honest that the lower-order term was found by fitting residuals rather than derived. The paper also states the conditional nature of Theorem 3.1 directly, which is more than many heuristic papers do.

The soft spots are load-bearing, not cosmetic. The theorem assumes that the Montgomery–Soundararajan average-order estimate (3.5) holds “in a similar form” for the modified singular series S_{q,0}, with no derivation and no precise statement of what changes when primes dividing q are excluded. Replacing A_{h,ℓ} by the leading term of (3.5) is exactly the step that could introduce q-dependent constants or error terms, and the proof does not address those. The error term in (3.5) also has exponent ℓ/2 − 1/7ℓ + ϵ, and the proof appears to discard it without checking uniformity over the ranges in question.

Lemma 4.3 is a second major gap. Gallagher’s result is quoted without saying whether it is unconditional or conditional on the Hardy-Littlewood conjecture, and the use of e^{−λ} with λ = c log₂ p_N pushes the bound far outside the range in which that exponential tail is known. Uniformity in n is not established. Since Lemma 4.3 is what lets the paper discard all h > M log y, a failure there would invalidate the truncation argument.

Minor point: the numerical lower-order term is a fitted value, not a prediction. The author says this clearly, so it is not a hidden flaw, but it means the data cannot serve as independent confirmation of the conjectured shape.

Who this is for: readers interested in the heuristic shape of the Lemke Oliver–Soundararajan conjecture, especially the lower-order terms, will find the data and the organizing framework useful. As a rigorous proof, it is not there. I would want either a proof of the analogue of (3.5) for S_{q,0}, or a precise conjecture stated separately, and a complete proof or clear citation of Lemma 4.3 with uniform constants. Failing that, the paper should be reframed as a heuristic with numerical evidence. I would not currently cite it as evidence for the truncation bound, though I might cite the 10^18 data with attribution to the author.

For peer review: I would not desk-reject it, because the idea and data are worth a referee’s time, but acceptance would require either the missing proof or an honest reframing as a heuristic contribution. Send it to a specialist, not to a broad journal.

Referee Report

6 major / 4 minor

Summary. The paper studies the distribution of consecutive prime sequences modulo a prime q, in the direction of the Lemke Oliver-Soundararajan conjectures. After setting up a modified singular series S_{q,0} and rewriting the heuristic asymptotic for D(a,b;y), the paper states Theorem 3.1, which asserts that the quantities S_∅, S_{0}, S_{h}, and S_{0,h}, and hence D_n and D_{≥n}, decay as O_n((log_2 y)^n/(log y)^{n/2-1}) under an assumed analogue of the Montgomery-Soundararajan estimate (3.5) for S_{q,0}. The proof uses a prime-gap lemma (Lemma 4.3) to discard large h, and the paper also fits lower-order terms numerically using data up to 10^18. The abstract claims that this begins a rigorous connection between the Hardy-Littlewood prime k-tuple conjecture and the conjectured prime-pattern asymptotic.

Significance. If Theorem 3.1 were proven under the stated hypothesis, it would provide a useful framework for controlling truncation errors in Conjecture 2.2 and would clarify which terms in the Lemke Oliver-Soundararajan heuristic are negligible. The decomposition into S_∅, S_{0}, S_{h}, S_{0,h}, the use of Lemma 4.4 to relate the relevant finite differences, and the extension of numerical data to x=10^18 are useful contributions. However, the paper does not prove the analogue of (3.5) that its central theorem requires; it relies on a lemma whose proof is explicitly labeled 'not fully rigorous'; and the claimed lower-order correction is obtained by curve-fitting rather than by derivation. The advertised rigorous connection is therefore not established, and the paper's main value is as a heuristic and computational exploration rather than as a proof.

major comments (6)

[Section 3, Theorem 3.1] Theorem 3.1 is explicitly conditional on the assumption that (3.5) 'holds in a similar form' for S_{q,0}, but the paper never states the precise form of this analogue, nor does it offer a proof. The definitions of S_{q,0} and S_0 differ by the restriction (t+a,q)=1 and by the omission of primes dividing q, so the Euler factors are genuinely different; the mean value, the implied constants, and the error term in (3.5) need not transfer. In the proof of S_∅, the sum A_{h,ℓ} is simply replaced by the leading term of (3.5) 'appealing to our conjectured form'. Since all four bounds in Theorem 3.1 depend on this substitution, the central claim is not derived from the Hardy-Littlewood conjecture or from the known Montgomery-Soundararajan result alone.
[Section 4, Lemma 4.3] Lemma 4.3 is the only mechanism used to discard the contribution of h > M log y in Section 3, but its proof is prefaced with 'We sketch the details of the proof here. Although not fully rigorous, we expect the key ingredients of the proof to be present.' The proof cites Gallagher's bound P[g_n > λ log p_n] < e^{-λ} without stating whether that bound is unconditional or conditional on the Hardy-Littlewood conjecture, and without justifying that it may be applied uniformly with λ = c log_2 p_N for all n ≤ N. Because this lemma controls the truncation error in Theorem 3.1, the claimed O_n((log_2 y)^n/(log y)^{n/2-1}) bound is not rigorously established.
[Section 3, proof of Theorem 3.1] The bounds for S_{0}, S_{h}, and S_{0,h} are not actually proved. After the detailed treatment of S_∅, the paper states 'We proceed to evaluate S_{0}, S_{h}, and S_{0,h} in an analogous manner' and then asserts, 'In loose terms, taking k derivatives of A_{h,ℓ} corresponds to adding a factor of (log y)^k to the denominator.' No rigorous derivation or explicit final bound is supplied for these three terms. Since D_{≥n}(a,b;y) is the sum S_∅ + S_{0} + S_{h} + S_{0,h}, the claimed bound for D_{≥n} is incomplete.
[Section 3, proof of S_∅, Case 2] The treatment of odd ℓ in the proof of S_∅ appears to bound the wrong sign. In (3.18), the summand contains (-z)^ℓ, which is negative for odd ℓ, and the displayed expression is '-C_max' times a positive geometric series, so this is a lower bound, not an upper bound, for the absolute value of the contribution. To deduce S_∅ = O_n((log_2 y)^n/(log y)^{n/2-1}) one needs an estimate for |S_∅|; a negative lower bound does not control the size. The argument would need to take absolute values of the error terms in (3.5) throughout, which is not what is written.
[Appendix A] The claimed lower-order term of size O((log_2 x)^2/(log x)^2) is obtained by fitting residuals with SageMath's find_fit function, and no predicted constant or residual diagnostic is reported. This makes the term a fitted quantity, not a prediction of the conjectural asymptotic. The introduction and Section 3 present the lower-order analysis as tightening the heuristic in [4], but the numerical fitting in Appendix A cannot substitute for a derivation from the singular-series expressions. The sampling justification based on Theorem 3.1, which itself depends on the unproved (3.5) analogue and Lemma 4.3, is also circular for the purpose of validating the conjecture.
[Section 3, truncation argument] The truncation of D(a,b;y) to h ≤ M log y and to |T| ≥ n is justified using the average-order estimate (3.5) for the sums defining A_{h,ℓ}, B_{h,ℓ}, C_{h,ℓ}, and D_{h,ℓ}. But these are exactly the sums of S_{q,0}(A∪T) that define the object D being truncated. No independent estimate for S_{q,0} on individual sets is given, so the argument assumes the same type of cancellation it is trying to establish. This circularity is load-bearing: if the assumed analogue of (3.5) fails, the entire truncation and the resulting theorem collapse.

minor comments (4)

[Appendix A] There is a typo in the sentence 'Note that (A.2) approximates the the ratio π(...)': 'the the' should be 'the'.
[Section 3, proof of Theorem 3.1] The text refers to 'Lemma 4.2, whose statement and proof can be found in Appendix 4', but Lemma 4.2 is stated and proved in Section 4, not in an appendix.
[Appendix B] Figures 1-5 are referenced and described but not displayed in the manuscript text, making it impossible to assess the claimed agreement between the data, the main term (A.1), and the fitted lower-order term.
[Throughout] The notation 'O_n' is defined in the introduction, but the subscripts are sometimes omitted or ambiguous (for example in the statement of Lemma 4.3). The paper would benefit from a consistent notation for the dependence of implied constants on q as well as on n.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the central theorem is an explicitly conditional statement, and the numerically fitted lower-order term is not load-bearing.

full rationale

The paper's main new result, Theorem 3.1, is explicitly labeled as conditional: it assumes that the Montgomery-Soundararajan average-order estimate (3.5) 'holds in a similar form for S_{q,0}.' The proof then applies that assumption directly to the sums A_{h,l} that define S_empty and the other pieces of D_n. This is a valid derivation from a stated hypothesis, not a circular one: S_{q,0} is not defined in terms of D_n, and the proof does not use D_n to establish the assumed average-order formula. The unproved analogue of (3.5) for S_{q,0} is a serious rigor gap in the advertised 'rigorous connection,' but it is a missing assumption, not a circular step. Lemma 4.3 is admittedly only a sketch and cites Gallagher without stating whether the large-prime-gap bound is unconditional or conditional on the Hardy-Littlewood conjecture; again, this undermines rigor but does not make the argument circular. The Appendix A lower-order term obtained with SageMath's find fit function is explicitly called 'possible' and 'plausible,' and it is not used in the proof of Theorem 3.1. No prediction in the paper is forced by construction from its own fitted inputs or self-citations.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

Central claim rests on two unproved analytic assumptions: the analog of (3.5) for S_{q,0}, and Gallagher's large-gap bound as stated in Lemma 4.3. The lower-order numerical term is fitted to data. No new entities are introduced.

free parameters (2)

lower-order coefficient in O((log_2 y)^2/(log y)^2) = not given
Appendix A: 'Using SageMath's find fit function suggested a possible lower order term of size O((log_2 x)^2/(log x)^2).' This is fitted to the extended data, not predicted.
constant c in M = c log_2 y = sufficiently large integer depending on n
Chosen by hand in Section 3 to discard the h > M log y portion of the sum. Not fitted to data, but a free choice in the derivation.

assumptions (4)

domain assumption Hardy-Littlewood prime k-tuple conjecture (Conjecture 2.1)
The conjectured asymptotics for pi(x;q,(a,b)) in Conjecture 2.2 are derived from this external conjecture.
ad hoc to paper Montgomery-Soundararajan average order (3.5) holds in a similar form for S_{q,0}
Explicit assumption in Theorem 3.1; no proof is given that the singular series modification preserves the growth rate.
ad hoc to paper Gallagher exponential large-gap bound as used in Lemma 4.3
The lemma is only sketched, and the paper does not state whether Gallagher's bound is unconditional or conditional on the Hardy-Littlewood conjecture. It is needed to discard h > M log y.
domain assumption Primality modeled as independent binomial events with probability 1/log x
Used in Appendix A to justify the 10^-4 precision of the sampling estimates for extended data.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Nonuniform Distributions of Residues of Prime Sequences in Prime Moduli." pith.science (2026). https://pith.science/paper/I6M7WTUS

@misc{pith2026190807095,
  author       = {Pith},
  title        = {Pith review of: Nonuniform Distributions of Residues of Prime Sequences in Prime Moduli},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6M7WTUS}},
  note         = {Machine review of arXiv:1908.07095}
}

abstract

For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $\varphi(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $\pi(x)/\varphi(q)^2$. We begin to rigorously connect the Hardy-Littlewood prime $k$-tuple conjecture to a conjectured asymptotic formula for the frequencies of prime patterns modulo $q$.

Figures

Figures reproduced from arXiv: 1908.07095 by the authors.

**Figure 3.** The extended data for (1, 2) modulo 3. The slight bump at 5 · 1010 is due to stitching the raw data and sampled data together [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 5.** The residuals when curve fitted terms of size O((log2 y) 2/(log y) 2 ) and (A.1) are subtracted from π(x; 3,(1, 1)) for x0 ≤ x ≤ x1 where π(x0) = 104 and π(x1) = 1018 , using the extended data. References [1] T. M. Apostol, Introduction to Analytic Number Theory. Springer, 1998. [2] P. G. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen o… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[4]

Unexpected biases in the distribution of con- secutive primes,

R. J. L. Oliver and K. Soundararajan, “Unexpected biases in the distribution of con- secutive primes,” 2016

work page 2016
[1]

T. M. Apostol, Introduction to Analytic Number Theory . Springer, 1998

work page 1998
[2]

P. G. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Diﬀerenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enth¨ alt, p. 342–359. Cambridge Library Collection - Math- ematics, Cambridge University Press, 2013

work page 2013
[3]

Sharper Bounds for the Chebyshev functions θ(x) and ψ(x). ii,

L. Schoenfeld, “Sharper Bounds for the Chebyshev functions θ(x) and ψ(x). ii,” Math- ematics of Computation , vol. 30, no. 134, pp. 337–360, 1976

work page 1976
[5]

Lettre de M. le Professeur Tch´ ebychev ` a M. Fuss sur un nouveaux th´ eor` eme relatif aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3,

“Lettre de M. le Professeur Tch´ ebychev ` a M. Fuss sur un nouveaux th´ eor` eme relatif aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3,” 1853

work page
[6]

Chebyshev’s bias,

M. Rubinstein and P. Sarnak, “Chebyshev’s bias,” Experiment. Math. , vol. 3, no. 3, pp. 173–197, 1994. 15

work page 1994
[7]

Strings of congruent primes,

D. K. Shiu, “Strings of congruent primes,” Journal of the London Mathematical Society, vol. 61, no. 2, pp. 359–373, 2000

work page 2000
[8]

Dense clusters of primes in subsets,

J. Maynard, “Dense clusters of primes in subsets,” Compositio Mathematica, vol. 152, no. 7, pp. 1517–1554, 2016

work page 2016

Show all 10 references

[9]

Primes in short intervals,

H. L. Montgomery and K. Soundararajan, “Primes in short intervals,” Communications in Mathematical Physics , 2004

2004
[10]

On the distribution of primes in short intervals,

P. Gallagher, “On the distribution of primes in short intervals,” Mathematika, vol. 23, no. 1, pp. 4–9, 1976. 16

1976

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [4]

Unexpected biases in the distribution of con- secutive primes,

R. J. L. Oliver and K. Soundararajan, “Unexpected biases in the distribution of con- secutive primes,” 2016

work page 2016

[2] [1]

T. M. Apostol, Introduction to Analytic Number Theory . Springer, 1998

work page 1998

[3] [2]

P. G. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Diﬀerenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enth¨ alt, p. 342–359. Cambridge Library Collection - Math- ematics, Cambridge University Press, 2013

work page 2013

[4] [3]

Sharper Bounds for the Chebyshev functions θ(x) and ψ(x). ii,

L. Schoenfeld, “Sharper Bounds for the Chebyshev functions θ(x) and ψ(x). ii,” Math- ematics of Computation , vol. 30, no. 134, pp. 337–360, 1976

work page 1976

[5] [5]

Lettre de M. le Professeur Tch´ ebychev ` a M. Fuss sur un nouveaux th´ eor` eme relatif aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3,

“Lettre de M. le Professeur Tch´ ebychev ` a M. Fuss sur un nouveaux th´ eor` eme relatif aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3,” 1853

work page

[6] [6]

Chebyshev’s bias,

M. Rubinstein and P. Sarnak, “Chebyshev’s bias,” Experiment. Math. , vol. 3, no. 3, pp. 173–197, 1994. 15

work page 1994

[7] [7]

Strings of congruent primes,

D. K. Shiu, “Strings of congruent primes,” Journal of the London Mathematical Society, vol. 61, no. 2, pp. 359–373, 2000

work page 2000

[8] [8]

Dense clusters of primes in subsets,

J. Maynard, “Dense clusters of primes in subsets,” Compositio Mathematica, vol. 152, no. 7, pp. 1517–1554, 2016

work page 2016