Large prime factors of well-distributed sequences

Abhishek Bharadwaj; Brad Rodgers

arxiv: 2402.11884 · v4 · submitted 2024-02-19 · 🧮 math.NT

Large prime factors of well-distributed sequences

Abhishek Bharadwaj , Brad Rodgers This is my paper

Pith reviewed 2026-05-24 04:11 UTC · model grok-4.3

classification 🧮 math.NT

keywords large prime factorsarithmetic sequencesPoisson-Dirichlet processlevel of distributionupper bound sieveshifted primespolynomial values

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The pith

If an arithmetic sequence has level of distribution 1, the large prime factors of its random elements tend to a Poisson-Dirichlet process.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the distribution of large prime factors for a random element u drawn from arithmetic sequences that meet basic regularity and equidistribution conditions. It shows that level of distribution 1 forces these factors to converge in distribution to a Poisson-Dirichlet process. With only positive level of distribution the correlation functions still match those of the process, provided the test functions have restricted support. The authors also prove that the probability the largest prime factor exceeds u to the power 1 minus epsilon is bounded by a constant times epsilon. The results cover sequences such as shifted primes and values taken by irreducible polynomials.

Core claim

For arithmetic sequences satisfying simple regularity and equidistribution properties, level of distribution 1 implies that the large prime factors of a random element u tend to a Poisson-Dirichlet process. The proof combines the Arratia-Kochman-Miller characterization of that process with an upper bound sieve. Any positive level of distribution is enough to make the correlation functions of the large prime factors converge to those of the Poisson-Dirichlet process against test functions of restricted support. The probability that the largest prime factor of u exceeds u to the power 1 minus epsilon is O(epsilon).

What carries the argument

The Poisson-Dirichlet process, obtained via the Arratia-Kochman-Miller characterization after applying an upper bound sieve to the arithmetic sequence.

If this is right

Level of distribution 1 forces convergence in distribution of the large prime factors to the Poisson-Dirichlet process.
Any positive level of distribution forces the correlation functions to match those of the process for test functions of restricted support.
The probability that the largest prime factor exceeds u^{1-ε} is O(ε).
The conclusions hold for shifted primes and for values of single-variable irreducible polynomials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sieve techniques could be applied to other statistics of the same sequences beyond prime factors.
The restriction to test functions of limited support for positive level of distribution suggests a natural next target is to remove that support limitation.

Load-bearing premise

The arithmetic sequence must satisfy the stated regularity and equidistribution properties and possess level of distribution 1.

What would settle it

Explicit computation of the joint distribution of the three largest prime factors (normalized by log log u) for many random elements u from a concrete sequence such as shifted primes, checked against the known moments of the Poisson-Dirichlet process.

read the original abstract

We study the distribution of large prime factors of a random element $u$ of arithmetic sequences satisfying simple regularity and equidistribution properties. We show that if such an arithmetic sequence has level of distribution $1$ the large prime factors of $u$ tend to a Poisson-Dirichlet process, while if the sequence has any positive level of distribution the correlation functions of large prime factors tend to a Poisson-Dirichlet process against test functions of restricted support. For sequences with positive level of distribution, we also estimate the probability the largest prime factor of $u$ is greater than $u^{1-\epsilon}$, showing that this probability is $O(\epsilon)$. Examples of sequences described include shifted primes and values of single-variable irreducible polynomials. The proofs involve (i) a characterization of the Poisson-Dirichlet process due to Arratia-Kochman-Miller and (ii) an upper bound sieve.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Applies Poisson-Dirichlet to shifted primes and polynomials via level-of-distribution but upper-bound sieve may not give the matching asymptotics needed for full convergence.

read the letter

The main point here is that sequences with level of distribution 1 have large prime factors whose joint distribution follows the Poisson-Dirichlet process, while any positive level gives the correlation functions against test functions of limited support. The paper also bounds the chance that the largest prime factor exceeds u to the 1-epsilon power by O(epsilon). The examples are shifted primes and values of irreducible polynomials in one variable. This is a direct extension of existing work on Poisson-Dirichlet statistics to these arithmetic progressions under the stated regularity and equidistribution assumptions. The proofs rest on the Arratia-Kochman-Miller characterization together with an upper bound sieve. That combination is standard, and the paper does a clean job of identifying which sequences meet the hypotheses and stating the conclusions without extra claims. The soft spot is exactly the one flagged in the stress test. Upper bound sieves deliver one-sided estimates, yet convergence in distribution to the full point process requires control on the intensities over finite collections of intervals. The abstract gives no sign that lower bounds or exact main terms are proved once the level reaches 1, so the step from sieve inequalities to the precise joint counts needed for the characterization is the part that needs the most scrutiny in the proofs. Minor gaps in error terms or support restrictions would be normal for this style of work, but this one sits at the center of the argument. The paper is written for analytic number theorists who already know sieve methods and probabilistic models for prime factors. A reader working on distribution in special sequences will get concrete new statements to check against known level-of-distribution results. It is worth sending to referees because the claims are precise, the tools are named, and the examples are natural; a careful review can settle whether the sieve application closes the gap.

Referee Report

1 major / 1 minor

Summary. The paper claims that for arithmetic sequences satisfying regularity and equidistribution properties, a level of distribution equal to 1 implies that the large prime factors of a random element u converge in distribution to a Poisson-Dirichlet process, while any positive level of distribution yields convergence of the correlation functions to those of the Poisson-Dirichlet process when tested against functions of restricted support. It further claims an O(ε) bound on the probability that the largest prime factor of u exceeds u^{1-ε}. The proofs are said to rest on the Arratia-Kochman-Miller characterization of the Poisson-Dirichlet process together with an upper bound sieve; examples include shifted primes and values of irreducible polynomials.

Significance. If the central claims hold, the work would extend Poisson-Dirichlet statistics for large prime factors from the classical setting of integers or primes to a broader family of well-distributed arithmetic sequences, with potential consequences for the study of prime factors of polynomial values and shifted primes. The explicit invocation of the Arratia-Kochman-Miller characterization and the sieve method provides a clear technical route, though the manuscript supplies no proof details or error analysis in the abstract.

major comments (1)

[Abstract] Abstract: the passage from an upper-bound sieve to the precise joint intensities required by the Arratia-Kochman-Miller characterization for convergence in distribution to the Poisson-Dirichlet process is not addressed. Upper-bound sieves (Brun or Selberg type) typically furnish one-sided estimates; matching lower bounds or asymptotic equivalences over all finite collections of intervals appear necessary for the process limit, yet the abstract gives no indication that such lower bounds are established once the level of distribution reaches 1.

minor comments (1)

[Abstract] The abstract refers to 'simple regularity and equidistribution properties' without listing them; an explicit statement of these hypotheses in the introduction would clarify the scope of the theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and constructive comments on the abstract. We address the concern point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the passage from an upper-bound sieve to the precise joint intensities required by the Arratia-Kochman-Miller characterization for convergence in distribution to the Poisson-Dirichlet process is not addressed. Upper-bound sieves (Brun or Selberg type) typically furnish one-sided estimates; matching lower bounds or asymptotic equivalences over all finite collections of intervals appear necessary for the process limit, yet the abstract gives no indication that such lower bounds are established once the level of distribution reaches 1.

Authors: The abstract is intentionally concise and omits technical details of the argument. In the body of the manuscript, the assumption of level of distribution 1 is used to derive both the upper estimates from the sieve and the matching lower bounds (via standard inclusion of the level-of-distribution hypothesis into the sieve weights), yielding the precise asymptotic equivalences for the joint intensities over finite collections of intervals that are required by the Arratia-Kochman-Miller characterization. We will revise the abstract to indicate that level of distribution 1 produces the necessary two-sided asymptotics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external inputs

full rationale

The paper derives its main claims about Poisson-Dirichlet convergence for sequences with level of distribution 1 (or positive level) by invoking the external Arratia-Kochman-Miller characterization together with a standard upper-bound sieve. No self-definitional steps, fitted parameters renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the abstract or described proof outline. The derivation chain is therefore self-contained against external mathematical benchmarks rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on domain assumptions about the equidistribution level of the sequences studied and on standard background results from sieve theory and the theory of the Poisson-Dirichlet process.

axioms (2)

domain assumption Arithmetic sequences satisfy regularity and equidistribution properties
Invoked to define the class of sequences to which the theorems apply.
domain assumption Level of distribution is 1 or positive
Central hypothesis required for the Poisson-Dirichlet convergence statements.

pith-pipeline@v0.9.0 · 5679 in / 1200 out tokens · 28299 ms · 2026-05-24T04:11:23.333001+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 1 internal anchor

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B. Feng and J. Wu. On the density of shifted primes with large prime factors.Sci. China Math.61 (2018), no. 1, 83–94

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´E. Fouvry. Th´ eor` eme de Brun-Titchmarsh: application au th´ eor` eme de Fermat.Invent. Math.79 (1985), no. 2, 383–407

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Friedlander and H

J. Friedlander and H. Iwaniec. Opera de cribro. American Mathematical Society Colloquium Publications, 57.American Mathematical Society, Providence, RI, 2010. xx+527 pp

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C. Hooley. On the largest prime factor ofp+a.Mathematika20 (1973), 135–143

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Hough, M

J.B. Hough, M. Krishnapur, Y. Peres, B. Vir´ ag. Zeros of Gaussian analytic functions and de- terminantal point processes. University Lecture Series, 51.American Mathematical Society, Providence, RI, 2009. x+154 pp

work page 2009
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Iwaniec and E

H. Iwaniec and E. Kowalski. Analytic number theory. American Mathematical Society Col- loquium Publications, 53.American Mathematical Society, Providence, RI, 2004. xii+615 pp

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J.F.C. Kingman. Random discrete distributions.J. Roy. Statist. Soc. Ser. B37 (1975), 1–22

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Knuth and L

D.E. Knuth and L. Trabb Pardo. Analysis of a simple factorization algorithm.Theoret. Comput. Sci.3 (1976/77), no. 3, 321–348

work page 1976
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E. S. Lee, Explicit Mertens’ theorems for number fields.Bull. Aust. Math. Soc.108 (2023), no. 1, 169–172

work page 2023
[27]

G. Martin. An asymptotic formula for the number of smooth values of a polynomial.J. Number Theory93 (2002), no. 2, 108–182

work page 2002
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Merikoski

J. Merikoski. On the largest prime factor ofn 2 + 1.J. Eur. Math. Soc. (JEMS)25 (2023), no. 4, 1253–1284

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H. L. Montgomery and R.C. Vaughan. Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97.Cambridge University Press, Cambridge,

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A. Mounier. Un crible minorant effectif pour les entiers friables.arXiv preprint 2402.13198 (2024)

work page arXiv 2024
[31]

T. Nagel. G´ en´ eralisation d’un th´ eor` eme de Tchebycheff,Journ. de Math., 8 (1921), no.4, 343–356

work page 1921
[32]

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I. Niven, H. Zuckerman, H.L. Montgomery. An introduction to the theory of numbers. Fifth edition.John Wiley & Sons, Inc., New York, 1991. xiv+529 pp

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[33]

Pomerance

C. Pomerance. Popular values of Euler’s function.Mathematika27 (1980), no. 1, 84–89

work page 1980
[34]

The level of distribution of the Thue-Morse sequence

Spiegelhofer, Lukas. The level of distribution of the Thue-Morse sequence. Compos. Math. 156 (2020), no. 12, 2560–2587

work page 2020
[35]

T. Tao. The Poisson-Dirichlet process, and large prime factors of a random number.https://terrytao.wordpress.com/2013/09/21/ the-Poisson-Dirichlet-process-and-large-prime-factors-of-a-random-number/

work page 2013
[36]

A.M. Vershik. Asymptotic distribution of decompositions of natural numbers into prime divisors. (Russian)Dokl. Akad. Nauk SSSR289 (1986), no. 2, 269–272

work page 1986
[37]

Z. Wang. Autour des plus grands facteurs premiers d’entiers cons´ ecutifs voisins d’un entier cribl´ e.Q. J. Math.69 (2018), no. 3, 995–1013

work page 2018
[38]

J. Wu. On shifted primes with large prime factors and their products.Arch. Math. (Basel) 112 (2019), no. 4, 387–393. Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam 603103, India Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada E-mail addresses:bharadwaj.work@outlook.com, brad.w.rod...

work page 2019

[1] [1]

Extensions of Billingsley's Theorem via Multi-Intensities

R. Arratia, F. Kochman, and V.S. Miller. Extensions of Billingsley’s Theorem via Multi- Intensities.arXiv preprint arXiv:1401.1555

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

Baker and G

R. Baker and G. Harman. The Brun-Titchmarsh theorem on average.Analytic number theory, Vol. 1 (Allerton Park, IL, 1995), 39–103, Progr. Math., 138,Birkh¨ auser Boston, Boston, MA, 1996

work page 1995

[3] [3]

Banks and I.E

W.D. Banks and I.E. Shparlinski. On values taken by the largest prime factor of shifted primes.J. Aust. Math. Soc.82 (2007), no. 1, 133–147

work page 2007

[4] [4]

Billingsley

P. Billingsley. On the distribution of large prime divisors.Period. Math. Hungar.2 (1972), 283–289

work page 1972

[5] [5]

Billingsley

P. Billingsley. Convergence of probability measures. Second edition. Wiley Series in Prob- ability and Statistics: Probability and Statistics. A Wiley-Interscience Publication.John Wiley & Sons, Inc., New York, 1999. x+277 pp

work page 1999

[6] [6]

Cojocaru and M.R

A.C. Cojocaru and M.R. Murty. An introduction to sieve methods and their applications. London Mathematical Society Student Texts, 66.Cambridge University Press, Cambridge,

work page

[7] [7]

Dartyge, G

C. Dartyge, G. Martin, and G. Tenenbaum. Polynomial values free of large prime factors. Period. Math. Hungar. 43 (2001), no. 1-2, 111–119

work page 2001

[8] [8]

de la Bret` eche and S

R. de la Bret` eche and S. Drappeau. Niveau de r´ epartition des polynˆ omes quadratiques et crible majorant pour les entiers friables.J. Eur. Math. Soc. (JEMS)22 (2020), no. 5, 1577– 1624

work page 2020

[9] [9]

Deshouillers and H

J.-M. Deshouillers and H. Iwaniec. On the greatest prime factor ofn 2 +1.Ann. Inst. Fourier (Grenoble)32 (1982), no. 4, 1–11 (1983)

work page 1982

[10] [10]

K. Dickman. On the frequency of numbers containing prime factors of a certain relative magnitude.Arkiv f¨ or Matematik, Astronomi och Fysik.(1930) 22A (10): 1–14

work page 1930

[11] [11]

Y. Ding. On a conjecture on shifted primes with large prime factors.Arch. Math. (Basel) 120 (2023), no. 3, 245–252

work page 2023

[12] [12]

Y. Ding. On a conjecture on shifted primes with large prime factors, II.arXiv preprint 2402.09829(2024)

work page arXiv 2024

[13] [13]

Feng and J

B. Feng and J. Wu. On the density of shifted primes with large prime factors.Sci. China Math.61 (2018), no. 1, 83–94

work page 2018

[14] [14]

P. Erd˝ os. On the normal number of prime factors ofp−1 and some related problems concerning Euler’sϕ-function.Q. J. Math.os-6 (1935) no. 1, 205–213

work page 1935

[15] [15]

´E. Fouvry. Th´ eor` eme de Brun-Titchmarsh: application au th´ eor` eme de Fermat.Invent. Math.79 (1985), no. 2, 383–407

work page 1985

[16] [16]

Friedlander and H

J. Friedlander and H. Iwaniec. Opera de cribro. American Mathematical Society Colloquium Publications, 57.American Mathematical Society, Providence, RI, 2010. xx+527 pp

work page 2010

[17] [17]

A.O. Gelfond. Sur les nombres qui ont des propri´ et´ es additives et multiplicatives donn´ ees. (French)Acta Arith.13 (1967/68), 259–265

work page 1967

[18] [18]

Goldfeld

M. Goldfeld. On the number of primespfor whichp+ahas a large prime factor.Mathematika 16 (1969), 23–27

work page 1969

[19] [19]

Granville

A. Granville. Smooth numbers: computational number theory and beyond.Algorithmic number theory: lattices, number fields, curves and cryptography,267–323, Math. Sci. Res. Inst. Publ., 44,Cambridge Univ. Press, Cambridge, 2008. LARGE PRIME FACTORS 15

work page 2008

[20] [20]

C. Hooley. On the greatest prime factor of a quadratic polynomial.Acta Math.117 (1967), 281–299

work page 1967

[21] [21]

C. Hooley. On the largest prime factor ofp+a.Mathematika20 (1973), 135–143

work page 1973

[22] [22]

Hough, M

J.B. Hough, M. Krishnapur, Y. Peres, B. Vir´ ag. Zeros of Gaussian analytic functions and de- terminantal point processes. University Lecture Series, 51.American Mathematical Society, Providence, RI, 2009. x+154 pp

work page 2009

[23] [23]

Iwaniec and E

H. Iwaniec and E. Kowalski. Analytic number theory. American Mathematical Society Col- loquium Publications, 53.American Mathematical Society, Providence, RI, 2004. xii+615 pp

work page 2004

[24] [24]

J.F.C. Kingman. Random discrete distributions.J. Roy. Statist. Soc. Ser. B37 (1975), 1–22

work page 1975

[25] [25]

Knuth and L

D.E. Knuth and L. Trabb Pardo. Analysis of a simple factorization algorithm.Theoret. Comput. Sci.3 (1976/77), no. 3, 321–348

work page 1976

[26] [26]

E. S. Lee, Explicit Mertens’ theorems for number fields.Bull. Aust. Math. Soc.108 (2023), no. 1, 169–172

work page 2023

[27] [27]

G. Martin. An asymptotic formula for the number of smooth values of a polynomial.J. Number Theory93 (2002), no. 2, 108–182

work page 2002

[28] [28]

Merikoski

J. Merikoski. On the largest prime factor ofn 2 + 1.J. Eur. Math. Soc. (JEMS)25 (2023), no. 4, 1253–1284

work page 2023

[29] [29]

H. L. Montgomery and R.C. Vaughan. Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97.Cambridge University Press, Cambridge,

work page

[30] [30]

A. Mounier. Un crible minorant effectif pour les entiers friables.arXiv preprint 2402.13198 (2024)

work page arXiv 2024

[31] [31]

T. Nagel. G´ en´ eralisation d’un th´ eor` eme de Tchebycheff,Journ. de Math., 8 (1921), no.4, 343–356

work page 1921

[32] [32]

Niven, H

I. Niven, H. Zuckerman, H.L. Montgomery. An introduction to the theory of numbers. Fifth edition.John Wiley & Sons, Inc., New York, 1991. xiv+529 pp

work page 1991

[33] [33]

Pomerance

C. Pomerance. Popular values of Euler’s function.Mathematika27 (1980), no. 1, 84–89

work page 1980

[34] [34]

The level of distribution of the Thue-Morse sequence

Spiegelhofer, Lukas. The level of distribution of the Thue-Morse sequence. Compos. Math. 156 (2020), no. 12, 2560–2587

work page 2020

[35] [35]

T. Tao. The Poisson-Dirichlet process, and large prime factors of a random number.https://terrytao.wordpress.com/2013/09/21/ the-Poisson-Dirichlet-process-and-large-prime-factors-of-a-random-number/

work page 2013

[36] [36]

A.M. Vershik. Asymptotic distribution of decompositions of natural numbers into prime divisors. (Russian)Dokl. Akad. Nauk SSSR289 (1986), no. 2, 269–272

work page 1986

[37] [37]

Z. Wang. Autour des plus grands facteurs premiers d’entiers cons´ ecutifs voisins d’un entier cribl´ e.Q. J. Math.69 (2018), no. 3, 995–1013

work page 2018

[38] [38]

J. Wu. On shifted primes with large prime factors and their products.Arch. Math. (Basel) 112 (2019), no. 4, 387–393. Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam 603103, India Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada E-mail addresses:bharadwaj.work@outlook.com, brad.w.rod...

work page 2019