Pith. sign in

REVIEW 3 major objections 3 minor 6 cited by

On the exponents of distribution of primes and smooth numbers

T0 review · 3 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read Primes and smooth numbers are equidistributed to moduli up to $x^{5/8-o(1)}$ without assuming Selberg's eigenvalue conjecture.

desk verdict Real new results at x^{5/8} level, but the whole gain is carried by the author's own unproved preprint [40]; the paper deserves refereeing with that dependency made explicit. read the letter →

arxiv 2505.00653 v2 pith:UIV2VG5J submitted 2025-05-01 math.NT

Alexandru Pascadi This is my paper · ORCID
classification math.NT MSC 11N1311N2511N3511F7211L05
keywords exponentofdistributionprimessmoothnumberstriply-well-factorableweightsexceptionalMaassformslargesieveSelbergeigenvalueconjectureKloostermansums
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

This paper proves that primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to $x^{5/8-o(1)}$, an exponent previously within reach only on the assumption of Selberg's eigenvalue conjecture. For primes the result concerns triply-well-factorable sieve weights, and for smooth numbers it holds for arbitrary bounded weights. The mechanism is a pair of large sieve inequalities for exceptional Maass forms that save a factor depending on the exceptional spectral parameter $\theta_f$, replacing the pointwise Kim--Sarnak bound $\theta_f \le 7/32$ with an on-average saving. If the central estimate holds up, the twin-prime upper-bound constant drops from $3.229$ to $3.203+o(1)$ and the consecutive-smooth-number exponent improves from $3/5$ to $5/8$. The qualitative point is that $5/8$ is now unconditional: proving Selberg's conjecture would not improve these results.

What carries the argument

The central object is the exceptional-spectrum large sieve inequality for additively structured sequences, restated as Proposition 3.4 from the author's earlier work [40, Theorem 3]. It gives a saving $Y=\max(1, NH/(|a|(H+L)L \min_i T_H(\alpha_i)))$ in the $X^{\theta_f}$ factor that measures the failure of Selberg's eigenvalue conjecture. The companion Proposition 3.5, due to Watt, gives a similar saving for multiplicative convolutions when averaged over a level $q$. The argument threads these through Deshouillers--Iwaniec-style multilinear Kloosterman-sum estimates (Propositions 3.7 and 3.8) and uses the identity $S(mr,fn;c)=S(fmr,n;c)$ to shift the extra divisor $f$ into the entry where the additive large sieve applies. Section 2.3 shows that the entire gap between the old exponent $66/107$ and the new $5/8$ is precisely this $Y$-saving.

What would settle it

One could test Proposition 3.4 directly: choose $N,L,H,q,\alpha_i$ as in the proposition, set $a_n$ to the additive-convolution sequence displayed there, and compute the exceptional-spectrum sum in Assumption 3.2 for $X$ near $\max(1,q/N)Y$. A violation of the bound would disprove the proposition and remove the basis for Theorems 1.3 and 1.5.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes Theorem 1.3 and Theorem 1.5: for $Q \le x^{5/8-\varepsilon}$, the weighted sum over primes in arithmetic progressions has error $\ll_{\varepsilon,A,a} x/(\log x)^A$ for triply-well-factorable weights and for upper-bound well-factorable linear sieve weights (the latter already at $Q \le x^{3/5-\varepsilon}$), and the corresponding absolute error for smooth numbers is $\ll \Psi(x,y)/(\log x)^A$. In words, both primes and smooth numbers have exponent of distribution $5/8-o(1)$, matching what was previously known only conditional on Selberg's conjecture. The unconditional best before this paper was $66/107-\varepsilon\approx0.6168$. The proof eliminates the conjecture by combining the author's large sieve inequality for additively structured sequences with Watt's large sieve inequality for multiplicative convolutions, and by a Kloosterman-sum manipulation that moves the complementary-divisor variable to the other entry so that the additive structure can be used.

Load-bearing premise

The whole advance rests on Proposition 3.4, a large sieve inequality for additively structured sequences that is cited from the author's own earlier preprint [40] and is not proved here; if its saving $Y$ is weaker than claimed, the exponent falls back from $5/8$ toward $66/107$.

Editorial extensions

If this is right

  • Triply-well-factorable primes get an unconditional level of distribution $x^{5/8-o(1)}$, with weighted Bombieri--Vinogradov error $x/(\log x)^A$.
  • Smooth numbers in arithmetic progressions are equidistributed to moduli $x^{5/8-o(1)}$ with error $\Psi(x,y)/(\log x)^A$, improving the previous $66/107$ exponent.
  • The twin-prime sieve constant improves from $3.229$ to $3.203+o(1)$ times the Hardy--Littlewood prediction.
  • For consecutive smooth numbers, the upper-bound exponent $1+5/8-\varepsilon$ replaces the previous $1+3/5-\varepsilon$ in $\ll x \varrho(u)^{1+5/8-\varepsilon}$.
  • Conditional and unconditional exponents now coincide, so Selberg's conjecture no longer offers a path to better constants in this class of sieve problems.

Reading between the lines

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

  • The quantitative identity between the old and new exponents in Section 2.3 suggests the result is fragile: any independent check or weakening of Proposition 3.4 would directly change the final $5/8$, making that inequality the natural target for scrutiny.
  • If the non-Archimedean analogue of the same large sieve saving is developed, the fixed-residue restriction $a \ll x^\varepsilon$ could relax toward $a \ll x^{1+\varepsilon}$ for well-factorable $a$, extending the results to Goldbach-type counts.
  • A large sieve inequality for mixed additive-multiplicative convolution sequences would likely sharpen the Deshouillers--Iwaniec mean-value threshold toward $MN \le T^{5/8}$, as the paper notes; that exponent appearing again indicates a common barrier.
  • The duality between the prime and smooth-number reductions suggests that an exponent improvement found for either problem will transfer to the other, so future work can focus on the shared Kloosterman-fraction sum rather than on separate sieve setups.
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.

Referee Report

3 major / 3 minor

Summary. The paper announces unconditional exponent-of-distribution results at level x^{5/8-o(1)}: Theorem 1.3(i) for primes with triply-well-factorable weights, Theorem 1.5 for smooth numbers with arbitrary weights, as well as improvements for upper-bound linear sieve weights and for smooth moduli, with applications to twin primes and consecutive smooth numbers. The proof combines Deshouillers-Iwaniec dispersion arguments with newer large sieve inequalities for exceptional Maass forms: the author's large sieve for additively structured sequences, restated as Proposition 3.4 from the author's preprint [40], and Watt's theorem for multiplicatively structured sequences, stated as Proposition 3.5. The manuscript gives detailed reductions in Sections 4 and 6, a self-contained factorization analysis for linear sieve weights in Section 5, and refinements for smooth moduli in Section 7. The claimed mechanism is that a saving in the exceptional-spectrum factor for additive-convolution coefficients upgrades the prior threshold from 66/107 to 5/8 without assuming Selberg's eigenvalue conjecture.

Significance. If the supporting machinery were fully established inside the manuscript, the results would be a genuine advance: the unconditional level x^{5/8-o(1)} for these weight classes matches, in the relevant settings, the level previously available only under Selberg's eigenvalue conjecture, and the twin-prime constant improves from 3.229 to 3.203. The paper is generally careful: the inequalities in Sections 4, 5, and 6 are explicit, the linear-sieve factorization section is self-contained and detailed, and the limitations are openly stated in Section 2.3. The significance is, however, directly conditional on Proposition 3.4, which is only restated from an unpublished same-author preprint; no independent verification is cited. Thus the paper's main advertised claim is currently not verifiable from the manuscript alone.

major comments (3)
  1. [§3, Proposition 3.4] Proposition 3.4 is the core new ingredient behind the exponent 5/8, but it is only a restatement of the author's preprint [40, Theorem 3] and is not proved in this manuscript; no independent verification is cited. The saving Y enters Assumption 3.2 and propagates through Lemma 4.1, Lemma 4.2, Proposition 4.4, Lemma 6.1, Lemma 6.2, and Proposition 6.3 to Theorems 1.3(i) and 1.5. As Section 2.3 makes explicit, replacing the previous L-infinity bound on the exceptional-spectrum factor by this saving is exactly what changes the threshold from Q < x^{(5-4θ)/(8-6θ)} to Q < x^{5/8}; if the true saving were only Y = 1, the conditions (4.5) and (6.9) would fail at Q = x^{5/8-o(1)} and the exponent would fall back to 66/107. The paper therefore does not currently provide a complete proof of its main claim.
  2. [§4, Lemma 4.2] The proof of Lemma 4.2 is the place where the complementary-divisor variable f is moved into the other Kloosterman entry, and the text itself states that working around the coprimality constraint is a nontrivial argument. In the written proof, however, this step is summarized rather than demonstrated: after the substitution fdq = n1 - n2, the restriction (f,c) = 1 is relaxed and Lemma 4.1 is applied to W7, but the intervening manipulation of the Kloosterman sums is not written out in sufficient detail for a reader to verify the claimed bound. Since Lemma 4.2 is load-bearing for Proposition 4.4 and hence for Theorem 1.3(i), this omitted verification should be supplied.
  3. [§3, Proposition 3.8] Proposition 3.8 is another new input, giving a bound for sums of Kloosterman sums with multiplicative convolutions, but its proof is deferred to a 'closely follow' adaptation of [39, Corollary 17] and [40, Corollary 17], with the exceptional-spectrum step replaced by Proposition 3.5. Because Proposition 3.8 is used in both Lemma 4.1 and Lemma 6.1, the same verifiability concern applies: the key separation of variables in (3.8) and the subsequent application of Proposition 3.5 should be written out, or the proposition should be proved in full. At present the reader cannot check the numerical constants or the range of validity of (3.7) without consulting two external preprints.
minor comments (3)
  1. [§1, Corollary 1.6] The displayed bound for consecutive smooth numbers appears to have a rendering issue: 'xϱ(u)1+5/8−ε' should be clarified as x ϱ(u)^{1+5/8-ε} or x ϱ(u)^{5/8-ε}; the intended exponent should be stated explicitly.
  2. [§5.2, Corollary 1.4] The final twin-prime constant 3.20254 is obtained by adapting a Mathematica file from [32], but the numerical sieve integrals are only reported in a table; archiving the computation or providing enough details to reproduce the table independently would improve reproducibility.
  3. [§4, Proposition 4.5] In the proof of Proposition 4.5, the definition of z0 contains a duplicated phrase 'z0 := z0 :='; this should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 5/8 exponent is obtained by applying separate large-sieve inequalities, not by assuming the target equidistribution.

full rationale

The derivation chain is not circular. Theorems 1.3, 1.5, 5.4, and 7.1 are obtained by substituting the saving Y from Proposition 3.4 (a restatement of the author's separate large-sieve theorem [40, Thm 3]) and Watt's Proposition 3.5 into the existing Maynard/Lichtman/Drappeau frameworks, then solving explicit inequalities such as (4.5), (4.15), (6.5), and (6.9). No parameter is fitted to the final exponent: the level Q = x^{5/8-o(1)} emerges algebraically from the displayed constraints, e.g. Q < x^{(5-5theta)/(8-8theta)} = x^{5/8} in Section 2.3. The applications (twin primes, consecutive smooth numbers) are consequences, not inputs. The only same-author load-bearing input is [40, Theorem 3], restated as Proposition 3.4; however, that theorem is an independent large-sieve statement about additively-structured sequences and exceptional Maass forms, with assumptions that do not include primes or smooth-number equidistribution. Thus the citation does not make the target result an input. The paper even explicitly flags the technical limitation in Section 2.3 ('we do not know how to prove a corresponding large sieve inequality...'), which is an honest statement of an open point, not an admission of circularity. Any concern about the unproved status or possible weakness of [40] is a correctness or verification issue, not circularity. No self-definitional, fitted-input-called-prediction, ansatz-smuggling, or renaming pattern is present; score 0.

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

The central claim rests on external spectral results, especially the author's [40, Theorem 3], plus standard sieve and automorphic machinery. There are no fitted parameters and no invented physical or mathematical entities. The main risk is the same-author preprint [40] that supplies the decisive saving.

assumptions (6)
  • domain assumption Exceptional large sieve inequality for additively structured sequences ([40, Theorem 3], restated as Prop 3.4) gives the Y-saving in Assumption 3.2.
    This is the new analytic input; the paper's unconditional 5/8 exponent is exactly the saving (Q^2/x)^{θ_f} it provides. It is not proved in this manuscript.
  • domain assumption Watt's large sieve inequality for multiplicatively structured sequences ([45, Theorem 2], restated as Prop 3.5) holds with X ≤ Q^2/(N_1^2 N_2).
    Used in Prop 3.8 to handle the f-variable in the primes case after moving it to the m-entry of Kloosterman sums; Section 4.
  • standard math Kim-Sarnak bound θ_max ≤ 7/32 for exceptional Maass forms.
    Used throughout instead of Selberg's conjecture, e.g., Lemma 4.2 takes θ:=7/32.
  • standard math Kuznetsov trace formula and Deshouillers-Iwaniec spectral large sieve estimates for general sequences (Prop 3.3, [10]).
    Framework for all Kloosterman sum bounds; Prop 3.7 and 3.8 rely on it.
  • standard math Siegel-Walfisz condition and Bombieri-Vinogradov theorem for small moduli.
    Used to handle moduli q≤x^{1/2-ε} and to decompose von Mangoldt via Heath-Brown identity, as in Proposition 4.5 proof.
  • standard math Weil bound for Kloosterman sums (Lemma 3.6).
    Used in the dispersion-method estimates, e.g., bounding principal frequency contributions in Lemma 4.2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the exponents of distribution of primes and smooth numbers." pith.science (2026). https://pith.science/paper/UIV2VG5J

@misc{pith2026250500653,
  author       = {Pith},
  title        = {Pith review of: On the exponents of distribution of primes and smooth numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UIV2VG5J}},
  note         = {Machine review of arXiv:2505.00653}
}
abstract

We show that both primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to $x^{5/8 - o(1)}$, using triply-well-factorable weights for the primes (we also get improvements for the well-factorable linear sieve weights). This completely eliminates the dependency on Selberg's eigenvalue conjecture in previous works of Lichtman and the author, which built in turn on results of Maynard and Drappeau. We rely on recent large sieve inequalities for exceptional Maass forms of the author for additively-structured sequences, and on a related result of Watt for multiplicatively-structured sequences. As applications, we prove refined upper bounds for the counts of twin primes and consecutive smooth numbers up to $x$.

Figures

Figures reproduced from arXiv: 2505.00653 by the authors.

Figure 1
Figure 1. Structure of proofs (arrows represent logical implications). 3. Preliminaries and the exceptional spectrum 3.1. Analytic and combinatorial notation. We use the standard asymptotic notation f = o(g), f = O(g), f ≪ g, f ≍ g, indicating dependencies of implicit constants through subscripts (e.g, f = Oε(g) means |f| ≤ Cε|g| for some Cε > 0 depending only on ε). Statements like f(x) ≪ x o(1)g(x) should be read as ∀ε > 0,… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 6 Pith papers

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

  1. Non-abelian amplification and bilinear forms with Kloosterman sums

    math.NT 2025-11 accept novelty 8.0 of 10

    New non-abelian amplification bounds bilinear Kloosterman sums for composite moduli, saving c^{-1/12} for products of two primes of similar size.

  2. Bilinear forms with Kloosterman sums via quadratic characters

    math.NT 2026-07 accept novelty 7.0 of 10

    Bilinear Kloosterman forms save c^{-1/32} at length √c for all moduli via a new link to quadratic character sums, improving prior prime and composite bounds.

  3. Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

    math.NT 2026-02 unverdicted novelty 7.0 of 10

    Refinements of Harman's sieve yield Bombieri-Vinogradov-type mean value theorems for primes to moduli up to x^{9/17} (bilinear) and x^{17/32} (trilinear), plus new almost-all-q bounds on pi(x;q,a).

  4. Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications

    math.NT 2026-08 conditional novelty 6.0 of 10

    A new convolution-type Bombieri-Vinogradov theorem with well-factorable weights improves the lower density for P^+(n)<P^+(n+1) from 0.280 to 0.296 and the upper density for P^+(p-1)>=p^c from (7/2)log(1/c) to a smaller S(c).

  5. An improvement on the largest prime factors of consecutive integers

    math.NT 2026-07 conditional novelty 6.0 of 10

    The asymptotic density of n with P^+(n) < P^+(n+1) is shown to be > 0.280, a new record.

  6. An update on the Linnik--Goldbach problem

    math.NT 2026-05 unverdicted novelty 5.0 of 10

    Under GRH the Linnik-Goldbach problem is solved with six powers of two; unconditionally more than 25 percent of odd integers are a prime plus a power of two.

Reference graph

Works this paper leans on

48 extracted references · 44 canonical work pages · cited by 6 Pith papers

  1. [40]

    Large sieve inequalities for exceptional maass forms and the greatest prime factor ofn2 + 1

    Alexandru Pascadi. Large sieve inequalities for exceptional maass forms and the greatest prime factor ofn2 + 1. Preprint, arXiv:2404.04239v2, 2025

    arXiv 2025

  2. [1]

    Uniform Titchmarsh divisor problems.Adv

    Edgar Assing, Valentin Blomer, and Junxian Li. Uniform Titchmarsh divisor problems.Adv. Math., 393:Paper No. 108076, 51, 2021

    work page 2021

  3. [2]

    The mean square of the product of the Riemann zeta-function with Dirichlet polynomials.J

    Sandro Bettin, Vorrapan Chandee, and Maksym Radziwiłł. The mean square of the product of the Riemann zeta-function with Dirichlet polynomials.J. Reine Angew. Math. , 729:51–79, 2017. ON THE EXPONENTS OF DISTRIBUTION OF PRIMES AND SMOOTH NUMBERS 41

    work page 2017

  4. [3]

    On the large sieve.Mathematika, 12:201–225, 1965

    Enrico Bombieri. On the large sieve.Mathematika, 12:201–225, 1965

    work page 1965

  5. [4]

    Friedlander, and Henryk Iwaniec

    Enrico Bombieri, John B. Friedlander, and Henryk Iwaniec. Primes in arithmetic progressions to large moduli. Acta Math., 156(3-4):203–251, 1986

    work page 1986

  6. [5]

    Friedlander, and Henryk Iwaniec

    Enrico Bombieri, John B. Friedlander, and Henryk Iwaniec. Primes in arithmetic progressions to large moduli. II. Math. Ann., 277(3):361–393, 1987

    work page 1987

  7. [6]

    Friedlander, and Henryk Iwaniec

    Enrico Bombieri, John B. Friedlander, and Henryk Iwaniec. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. , 2(2):215–224, 1989

    work page 1989

  8. [7]

    Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables.J

    Régis de la Bretèche and Sary Drappeau. Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables.J. Eur. Math. Soc. , 22(5):1577–1624, 2020

    work page 2020

Show all 48 references
  1. [8]

    Deshouillers and H

    J.-M. Deshouillers and H. Iwaniec. On the greatest prime factor of n2 + 1. Ann. Inst. Fourier (Grenoble) , 32(4):1–11, 1982

    1982

  2. [9]

    Deshouillers and H

    J.-M. Deshouillers and H. Iwaniec. Power mean-values for Dirichlet’s polynomials and the Riemann zeta-function. II. Acta Arith., 43(3):305–312, 1984

    1984

  3. [10]

    Kloosterman sums and Fourier coefficients of cusp forms.Invent

    Jean-Marc Deshouillers and Henryk Iwaniec. Kloosterman sums and Fourier coefficients of cusp forms.Invent. Math., 70(2):219–288, 1982

    1982

  4. [11]

    Théorèmes de type Fouvry-Iwaniec pour les entiers friables.Compos

    Sary Drappeau. Théorèmes de type Fouvry-Iwaniec pour les entiers friables.Compos. Math., 151(5):828–862, 2015

    2015

  5. [12]

    Smooth-supported multiplicative functions in arith- metic progressions beyond thex1/2-barrier

    Sary Drappeau, Andrew Granville, and Xuancheng Shao. Smooth-supported multiplicative functions in arith- metic progressions beyond thex1/2-barrier. Mathematika, 63(3):895–918, 2017

    2017

  6. [13]

    Elliott and Heini Halberstam

    Peter D.T.A. Elliott and Heini Halberstam. A conjecture in prime number theory. InSymposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) , pages 59–72. Academic Press, London-New York, 1968

    1968

  7. [14]

    Répartition des suites dans les progressions arithmétiques.Acta Arith., 41(4):359–382, 1982

    Étienne Fouvry. Répartition des suites dans les progressions arithmétiques.Acta Arith., 41(4):359–382, 1982

    1982

  8. [15]

    Autour du théorème de Bombieri-Vinogradov.Acta Math., 152(3-4):219–244, 1984

    Étienne Fouvry. Autour du théorème de Bombieri-Vinogradov.Acta Math., 152(3-4):219–244, 1984

    1984

  9. [16]

    Sur le problème des diviseurs de Titchmarsh.J

    Étienne Fouvry. Sur le problème des diviseurs de Titchmarsh.J. Reine Angew. Math. , 357:51–76, 1985

    1985

  10. [17]

    Autour du théorème de Bombieri-Vinogradov

    Étienne Fouvry. Autour du théorème de Bombieri-Vinogradov. II.Ann. Sci. École Norm. Sup. (4) , 20(4):617–640, 1987

    1987

  11. [18]

    On a theorem of Bombieri-Vinogradov type.Mathematika, 27(2):135–152, 1980

    Étienne Fouvry and Henryk Iwaniec. On a theorem of Bombieri-Vinogradov type.Mathematika, 27(2):135–152, 1980

    1980

  12. [19]

    Primes in arithmetic progressions.Acta Arith., 42(2):197–218, 1983

    Étienne Fouvry and Henryk Iwaniec. Primes in arithmetic progressions.Acta Arith., 42(2):197–218, 1983

    1983

  13. [20]

    Entiers sans grand facteur premier en progressions arithmetiques.Proc

    Étienne Fouvry and Gérald Tenenbaum. Entiers sans grand facteur premier en progressions arithmetiques.Proc. London Math. Soc. (3) , 63(3):449–494, 1991

    1991

  14. [21]

    Répartition statistique des entiers sans grand facteur premier dans les progressions arithmétiques

    Étienne Fouvry and Gérald Tenenbaum. Répartition statistique des entiers sans grand facteur premier dans les progressions arithmétiques. Proc. London Math. Soc. (3) , 72(3):481–514, 1996

    1996

  15. [22]

    American Mathematical Society, Providence, RI, 2010

    John Friedlander and Henryk Iwaniec.Opera de cribro, volume 57 ofAmerican Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2010

    2010

  16. [23]

    Integers, without large prime factors, in arithmetic progressions

    Andrew Granville. Integers, without large prime factors, in arithmetic progressions. I.Acta Math., 170(2):255– 273, 1993

    1993

  17. [24]

    Integers, without large prime factors, in arithmetic progressions

    Andrew Granville. Integers, without large prime factors, in arithmetic progressions. II.Philos. Trans. Roy. Soc. London Ser. A , 345(1676):349–362, 1993

    1993

  18. [25]

    G. H. Hardy and J. E. Littlewood. Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes.Acta Math., 44(1):1–70, 1923

    1923

  19. [26]

    Adam J. Harper. Bombieri–Vinogradov and Barban–Davenport–Halberstam type theorems for smooth numbers. arXiv preprint arXiv:1208.5992 , 2012

    2012 arXiv

  20. [27]

    On the number of positive integers≤ x and free of prime factors> y

    Adolf Hildebrand. On the number of positive integers≤ x and free of prime factors> y. J. Number Theory , 22(3):289–307, 1986

    1986

  21. [28]

    A new form of the error term in the linear sieve.Acta Arith., 37:307–320, 1980

    Henryk Iwaniec. A new form of the error term in the linear sieve.Acta Arith., 37:307–320, 1980

    1980

  22. [29]

    American Mathematical Society, Providence, RI, 2021

    Henryk Iwaniec and Emmanuel Kowalski.Analytic number theory, volume 53. American Mathematical Society, Providence, RI, 2021

    2021

  23. [30]

    Henry H. Kim. Functoriality for the exterior square ofGL4 and the symmetric fourth ofGL2. J. Amer. Math. Soc., 16(1):139–183, 2003. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak

    2003

  24. [31]

    Kuznetsov

    Nikolai V. Kuznetsov. The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums.Mat. Sb. (N.S.) , 111(153)(3):334–383, 479, 1980

    1980

  25. [32]

    Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier

    Jared Duker Lichtman. Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier. Preprint, arXiv:2309.08522v1, 2023

    2023 arXiv

  26. [33]

    A modification of the linear sieve, and the count of twin primes.Algebra Number Theory, 19(1):1–38, 2025

    Jared Duker Lichtman. A modification of the linear sieve, and the count of twin primes.Algebra Number Theory, 19(1):1–38, 2025

    2025

  27. [34]

    Ju.˜V. Linnik. The dispersion method in binary additive problems . American Mathematical Society, Providence, RI, 1963. Translated by S. Schuur

    1963

  28. [35]

    Small gaps between primes.Ann

    James Maynard. Small gaps between primes.Ann. of Math. (2) , 181(1):383–413, 2015. 42 ALEXANDRU PASCADI

    2015

  29. [36]

    Primes in Arithmetic Progressions to Large Moduli I: Fixed Residue Classes.Mem

    James Maynard. Primes in Arithmetic Progressions to Large Moduli I: Fixed Residue Classes.Mem. Amer. Math. Soc., 306(1542), 2025

    2025

  30. [37]

    Primes in Arithmetic Progressions to Large Moduli II: Well-Factorable Estimates.Mem

    James Maynard. Primes in Arithmetic Progressions to Large Moduli II: Well-Factorable Estimates.Mem. Amer. Math. Soc., 306(1543), 2025

    2025

  31. [38]

    Primes in Arithmetic Progressions to Large Moduli III: Uniform Residue Classes.Mem

    James Maynard. Primes in Arithmetic Progressions to Large Moduli III: Uniform Residue Classes.Mem. Amer. Math. Soc., 306(1544), 2025

    2025

  32. [39]

    Smooth numbers in arithmetic progressions to large moduli

    Alexandru Pascadi. Smooth numbers in arithmetic progressions to large moduli. Compos. Math., to appear. Preprint, arXiv:2304.11696, 2023

    2023

  33. [41]

    D. H. J. Polymath. Variants of the Selberg sieve, and bounded intervals containing many primes.Res. Math. Sci., 1:Art. 12, 83, 2014

    2014

  34. [42]

    On the estimation of Fourier coefficients of modular forms

    Atle Selberg. On the estimation of Fourier coefficients of modular forms. InProc. Sympos. Pure Math., Vol. VIII, pages 1–15. Amer. Math. Soc., Providence, RI, 1965

    1965

  35. [43]

    On primes in arithmetic progressions and bounded gaps between many primes.Adv

    Julia Stadlmann. On primes in arithmetic progressions and bounded gaps between many primes.Adv. Math., 468:Paper No. 110190, 2025

    2025

  36. [44]

    A. I. Vinogradov. The density hypothesis for DirichetL-series. Izv. Akad. Nauk SSSR Ser. Mat. , 29:903–934, 1965

    1965

  37. [45]

    N. Watt. Kloosterman sums and a mean value for Dirichlet polynomials.J. Number Theory, 53(1):179–210, 1995

    1995

  38. [46]

    über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen

    Dieter Wolke. über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen. I.Math. Ann., 202:1–25, 1973

    1973

  39. [47]

    über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen

    Dieter Wolke. über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen. II.Math. Ann., 204:145–153, 1973

    1973

  40. [48]

    Bounded gaps between primes.Ann

    Yitang Zhang. Bounded gaps between primes.Ann. of Math. (2) , 179(3):1121–1174, 2014. Mathematical Institute, Radcliffe Obser v atory quarter, Woodstock Road, Oxford OX2 6GG, Eng- land Email address: alexpascadi@gmail.com

    2014

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.