Central limit theorems for random multiplicative functions over function fields

Asif Zaman; Declan Hoban; Jibran Iqbal Shah; Nadya-Catherine Ismail; William Verreault

REVIEW 1 major objections 3 minor 38 references

Central limit theorems for random multiplicative functions over function fields

T0 review · 1 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proves that a subset of polynomials over F_q[t] whose multiplicative energy is asymptotically trivial has Gaussian-distributed Steinhaus random multiplicative sums, and derives four central limit theorems from it.

desk verdict Solid function-field extension of Soundararajan–Xu with a real but likely repairable gap in the shifted-primes section. read the letter →

arxiv 2511.22905 v2 pith:CQVUH66C submitted 2025-11-28 math.NT math.PR

Declan Hoban , Jibran Iqbal Shah , Nadya-Catherine Ismail , William Verreault , Asif Zaman This is my paper · ORCID

classification math.NTmath.PR MSC 11T5560F0511K65

keywords randommultiplicativefunctionsfunctionfieldscentrallimittheoremenergyshortintervalsshiftedprimesroughpolynomialssievemethods

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's goal is to say when a randomly weighted sum over a set of monic polynomials of fixed degree over F_q[t] has a Gaussian limit. The main theorem gives a sufficient condition: if a set A is not too small and contains a nearly full subset S whose multiplicative energy is asymptotically trivial — meaning almost all solutions of F1F2=G1G2 come from swapping the two factors — then the normalized Steinhaus random multiplicative sum over A converges in distribution to the standard complex normal distribution CN(0,1). A sympathetic reader should care because this transfers a known integer-number-theory criterion to the function-field setting, where far fewer examples were known. The paper then checks the condition in four settings: short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. Along the way it establishes uniform estimates for smooth and rough polynomials in short intervals and a short-interval bound for multiplicative functions that are likely useful independently.

What carries the argument

The multiplicative energy E×(S) and a refined martingale filtration. The paper orders monic irreducible polynomials by degree and defines, for each prime P, the fiber S_P of polynomials in S whose ≺-maximal prime factor is P. Because the random value f(P) is independent of the sigma-algebra generated by earlier primes, the normalized sums over these fibers form a martingale difference sequence. A new bound on smooth polynomials in short intervals shows each fiber is sparse, which permits the size assumption |A| ≫ q^N exp(−(1/3)√(N log q)). The energy condition then forces the error terms in the martingale CLT to vanish.

What would settle it

Take q=3, N=2, Z=t, A=t, B=1, and the degree-one prime P=t. Write X={ (GA−Z)(GB−Z) : G monic linear }. For every G∈M_1, P divides GA−Z = t(G−1), so all 3 elements of X are divisible by P. The sieve local density used in the proof of Theorem 2.3 would predict only |X|/|P| = 1 such G. This direct count shows the shifted-prime bound (9.3) is not supported as written.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for any family of subsets A of monic degree-N polynomials over F_q[t] with |A| ⩾ q^N exp(−(1/3)√(N log q)), if there is S⊆A with |S|=(1+o(1))|A| and multiplicative energy E×(S)=|{(F1,F2,G1,G2)∈S^4 : F1F2=G1G2}|=(2+o(1))|S|^2 as q^N→∞, then (1/√|A|)∑_{F∈A} f(F) converges to CN(0,1) for a Steinhaus random multiplicative function f. The proof realizes the partial sum as a martingale difference sequence indexed by primes, using a refined filtration by the maximal prime factor under a degree-respecting ordering, and then applies a martingale central limit theorem; the energy condition controls the fourth-moment and variance-fluctuation terms. The paper derives fo

Load-bearing premise

The shifted-prime application assumes that in the sieve, the number of G for which D divides (GA−Z)(GB−Z) is exactly q^d/g(D) with g(P)=|P|/2 or |P| according as P divides AB(A−B), but this local density is not correct when P divides the shift Z and exactly one of A,B, so the resulting bound (9.3) is not established as written.

Editorial extensions

If this is right

If Theorem 1.1 is correct, any subset of monic degree-N polynomials satisfying the size and trivial-energy conditions automatically has a Gaussian limit for Steinhaus random multiplicative sums.
The short-interval application gives a CLT when q^h→∞ with q^{h+1}=o(q^N/N), and when h→∞ with q^{h+1}=o(q^N/N^c) for c>2 log 2−1.
The few-prime-factors application covers polynomials with k=o(log N) irreducible factors, matching the known integer range.
The shifted-prime and rough-polynomial applications give CLTs for sets of primes shifted by a fixed polynomial and for polynomials with all prime factors of degree exceeding z, with z≫√N.
The auxiliary estimates — smooth polynomials in short intervals, a short-interval bound for multiplicative functions, and a Chebyshev-type bound for rough polynomials — are uniform in q and N and may be used independently.

Reading between the lines

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

The energy criterion is transferable: any future family of polynomials with provably trivial multiplicative energy inherits the Gaussian conclusion without reworking the martingale proof.
The new short-interval estimates for smooth and rough polynomials may sharpen other function-field distribution questions, such as prime polynomial counts or divisor sums in intervals.
One could test the boundary of the short-interval CLT by computing the multiplicative energy of intervals with h between N/2 and N; the paper's bound is near-optimal in the fixed-degree, large-q limit.

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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

Colleague,

The short version: this is a meaningful extension of the Soundararajan–Xu CLT criterion to F_q[t], and most of it holds up. But the shifted-primes application (Theorem 2.3) has a genuine gap in its Selberg sieve step, so that theorem is not established as written.

What's actually new: the refined prime-wise martingale filtration (Lemma 5.2) lets them weaken the density condition from q^N/N to q^N exp(-(1/3)√(N log q)), and the auxiliary estimates — Proposition 2.5, Theorem 2.6, Lemma 2.7 — look genuinely useful. The main theorem and the short-interval, few-prime-factor, and rough-polynomial applications are coherent and carefully argued. This is not a routine translation; the filtration is a real technical improvement.

The soft spot is in §9. The local density claim for the shifted-prime sieve is false for primes dividing the shift Z. The paper says that for each P|D, the congruence (GA−Z)(GB−Z) ≡ 0 mod P leaves 2 residue classes when P ∤ AB(A−B) and 1 when P | AB(A−B). That ignores P | Z. If P | Z and P ∤ AB, both congruences force G ≡ 0 mod P — one class, not two. If P | Z and P | A, the first congruence holds for every G — all classes, not one. Concrete failure: q=3, Z=t, A=t^r+1, B=t^r+2, D=t. Then |X_D| = q^{d−1}, whereas the paper's formula with g(t)=|t|/2 gives 2q^{d−1}. So the remainder R_D in Lemma 3.2 is not zero, and (9.3) is unjustified. Since (9.3) drives the bound on Ξ, Theorem 2.3 lacks support. This looks fixable — split off primes dividing Z or use a different sifting set — but as written it is a real gap. Minor: in §7 the line 'using our choice m=(1+ε)N' should read m=(1+ε) log N; harmless typo.

The main theorem itself holds up on inspection. The citation pattern is fine; [2] has overlapping authors but is used as a lemma, not as the target result.

Bottom line: worth a serious referee. The paper deserves revision, not rejection. Send it to peer review with a request to repair the shifted-prime sieve argument; the rest of the package can be evaluated in the same round.

Referee Report

1 major / 3 minor

Summary. The paper proves a function-field analogue of the Soundararajan–Xu criterion: if A⊂M_N has size |A| ≫ q^N exp(-(1/3)√(N log q)) and contains a subset S with |S|=(1+o(1))|A| and asymptotically trivial multiplicative energy, then normalized sums of Steinhaus random multiplicative functions over A converge in distribution to CN(0,1). The proof uses a refined prime-based martingale filtration and a new estimate for smooth polynomials in short intervals. The criterion is applied to four families: polynomials in short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. Auxiliary results include an explicit Hildebrand-type inequality, a function-field Shiu theorem, and a short-interval Chebyshev bound.

Significance. If correct, Theorem 1.1 is a valuable and flexible criterion, extending a recent number-field theorem to F_q[t] and yielding four new CLTs. The paper is largely self-contained and provides explicit parameter-free estimates (Proposition 2.5, Theorem 2.6, Lemma 2.7) that should be of independent interest. The martingale argument for the main criterion is coherent. However, the shifted-prime application currently rests on a false Selberg-sieve local-density assertion, so Theorem 2.3 is not established as written. The remaining applications appear coherent and carefully argued.

major comments (1)

[§5.3, proof of Theorem 5.3] The moment computations are written with the wrong orthogonality relation. The displayed fourth moment is Σ E[f(F1)f(F2)f(G1)f(G2)] without conjugates, and the text says the expectation vanishes unless F1G1=F2G2. For |Z_P|^4 the correct expression contains $\overline{f(G_1)}\,\overline{f(G_2)}$, and the non-vanishing condition is F1F2=G1G2, matching (1.4) and condition (ii) of Theorem 5.3. The same inconsistency appears in the second-moment-squared argument. As written, the claimed bound on Σ E|Z_P|^4 and hence the proof of Theorem 1.1 does not follow. This is straightforward to repair, but it must be corrected systematically throughout the proof.

minor comments (3)

[§9] After the local-density correction, the condition 'P∤AB(A−B)' should be replaced by a condition involving Z as well, and the role of Z in the Pollack divisor M=AB(B−A)Z should be stated before the local-density claim, not only in the denominator computation.
[§5.3] The two occurrences of 'F1G1=F2G2' should read 'F1F2=G1G2'.
[§7, proof of Theorem 2.1(a)] In the final line of case (a), 'This verifies (iii)' appears to mean condition (ii) of Theorem 5.3; condition (iii) was already verified above.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main CLT criterion and applications are derived from independent martingale and sieve inputs; the few self-citations are used as lemmas, not as the source of the central claim.

full rationale

The paper's central result, Theorem 1.1, is a sufficient criterion rather than a fitted prediction: it takes as hypotheses a lower bound on |A| and asymptotically trivial multiplicative energy E×(S), and proves convergence to CN(0,1) via McLeish's martingale CLT and Lemma 5.2. The energy condition (1.4) is an input, not an output of the Gaussian conclusion, and the applications verify it by direct counting of solutions to F1F2 = G1G2. Lemma 5.2 is proved from Proposition 2.5, Lemma 4.1, and Gorodetsky's uniform smooth-polynomial estimate, none of which encode the CLT. The four applications each reduce Theorem 1.1 to independent number-theoretic estimates: short-interval energy (Proposition 7.1), few-prime-factor energy via combinatorial bounds including [2, Lemma 5], shifted-prime Selberg sieve estimates, and rough-polynomial energy via Lemma 4.4. The self-citations with overlapping authors ([2], [33]) are used as published lemmas or as context for negative examples, and the paper does not invoke a self-cited uniqueness theorem to force its choice of filtration. The skeptic's concern about §9 is a potential mathematical gap in the Selberg-sieve local densities when a prime divides the shift Z, but a false local density is an error of proof, not a circular reduction of the conclusion to its assumptions. Accordingly, the circularity score is low.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical/mathematical entities. It relies on standard analytic-number-theory results and on published estimates, several of which come from overlapping author groups but are used as external lemmas.

assumptions (8)

domain assumption Steinhaus random multiplicative function model: independent uniform unit-circle phases on primes, multiplicatively extended
Used throughout; this is the object of study and the source of the random sums.
standard math McLeish complex martingale central limit theorem
Theorem 5.1 is the engine of Theorem 5.3 and Theorem 1.1.
standard math Selberg sieve for polynomial rings (Webb)
Used for the rough-polynomial bound and shifted primes; Lemma 3.2 is quoted without proof.
standard math Function-field prime number theorem bounds π_q(n) ≈ q^n/n
Used pervasively in estimates involving primes, smooth polynomials, and rough polynomials.
domain assumption Gorodetsky uniform smooth polynomial estimate (Theorem 4.2)
External result on Ψ(N,d) used in the Shiu proof and Lemma 5.2.
domain assumption Elboim–Gorodetsky, Afshar–Porritt, and Hwang estimates for |P_k(N)|, |S_k(N)|, and |D_k(N)|
External estimates used to verify size and energy conditions in Theorem 2.2.
domain assumption Limit convention: q^N → ∞ allows q and N to vary arbitrarily (Remark 1.1.1)
All asymptotic claims are taken in this two-parameter limit; validity depends on the estimates being uniform in q.
standard math Random phase expectation E[f(P)^m]=0 for m≥1 and orthogonality of distinct primes
Used in the fourth-moment computations and characteristic function estimates.

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Pith. "Pith review of Central limit theorems for random multiplicative functions over function fields." pith.science (2026). https://pith.science/paper/CQVUH66C

@misc{pith2026251122905,
  author       = {Pith},
  title        = {Pith review of: Central limit theorems for random multiplicative functions over function fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CQVUH66C}},
  note         = {Machine review of arXiv:2511.22905}
}

abstract

We provide a sufficient characterization for subsets $\mathcal{A}$ of the polynomial ring $\mathbb{F}_q[t]$ for which partial sums of Steinhaus random multiplicative functions approach a complex standard normal distribution. This extends recent work of Soundararajan and Xu to the function field setting. We apply this characterization to deduce central limit theorems in four cases: polynomials in short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. In doing so, we also establish an explicit Hildebrand inequality for smooth polynomials in short intervals, a function field form of Shiu's theorem for multiplicative functions, and an explicit Chebyshev bound for rough polynomials in short intervals.

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Works this paper leans on

38 extracted references · 3 linked inside Pith

[2]

Sums of random multiplicative functions over function fields with few irreducible factors

D. Aggarwal et al. “Sums of random multiplicative functions over function fields with few irreducible factors”.Math. Proc. Cambridge Philos. Soc.173.3 (2022), pp. 715–726

2022
[1]

The function field Sathe-Selberg formula in arithmetic progressions and ‘short intervals’

A. Afshar and S. Porritt. “The function field Sathe-Selberg formula in arithmetic progressions and ‘short intervals’”.Acta Arith.187.2 (2019), pp. 101–124

2019
[3]

Atherfold and J

C. Atherfold and J. Najnudel.The Fourier coefficients of the critical holomorphic multiplicative chaos. https://arxiv.org/abs/2508.13849. 2025

arXiv 2025
[4]

Prime polynomials in short intervals and in arithmetic progressions

E. Bank, L. Bary-Soroker, and L. Rosenzweig. “Prime polynomials in short intervals and in arithmetic progressions”.Duke Math. J.164.2 (2015), pp. 277–295

2015
[5]

Bary-Soroker and N

L. Bary-Soroker and N. Goldgraber.Product Sets of Arithmetic Progressions in Function Fields.https: //arxiv.org/abs/2309.05046. 2023

arXiv 2023
[6]

Random multiplicative functions in short intervals

S. Chatterjee and K. Soundararajan. “Random multiplicative functions in short intervals”.Int. Math. Res. Not. IMRN2012.3 (2012), pp. 479–492

2012
[7]

Uniform estimates for almost primes over finite fields

D. Elboim and O. Gorodetsky. “Uniform estimates for almost primes over finite fields”.Proc. Amer. Math. Soc.150.7 (2022), pp. 2807–2822

2022
[8]

Gorodetsky and M

O. Gorodetsky and M. D. Wong.Multiplicative chaos measure for multiplicative functions: theL 1- regime.https://arxiv.org/abs/2503.10555. 2025

arXiv 2025

Show all 38 references

[9]

Gorodetsky and M

O. Gorodetsky and M. D. Wong.On the limiting distribution of sums of random multiplicative func- tions.https://arxiv.org/abs/2508.12956. 2025

2025
[10]

Uniform estimates for smooth polynomials over finite fields

O. Gorodetsky. “Uniform estimates for smooth polynomials over finite fields”.Discrete Anal.(2023), Paper No. 16, 31

2023
[11]

Smooth numbers: computational number theory and beyond

A. Granville. “Smooth numbers: computational number theory and beyond”.Algorithmic number the- ory: lattices, number fields, curves and cryptography. Vol. 44. Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge, 2008, pp. 267–323

2008
[12]

Mean values of multiplicative functions over function fields

A. Granville, A. J. Harper, and K. Soundararajan. “Mean values of multiplicative functions over function fields”.Res. Number Theory1 (2015), Paper No. 25, 18

2015
[13]

Gut.Probability: a graduate course

A. Gut.Probability: a graduate course. Second. Springer Texts in Statistics. Springer, New York, 2013

2013
[14]

A. J. Harper.Moments of random multiplicative functions, III: A short review.https://arxiv.org/ abs/2410.11523. 2024

2024 arXiv
[15]

Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos

A. J. Harper. “Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos”.Forum Math. Pi8 (2020), e1, 95

2020
[16]

On the limit distributions of some sums of a random multiplicative function

A. J. Harper. “On the limit distributions of some sums of a random multiplicative function”.J. Reine Angew. Math.678 (2013), pp. 95–124

2013
[17]

Integers free of large prime divisors in short intervals

A. Hildebrand. “Integers free of large prime divisors in short intervals”.Quart. J. Math. Oxford Ser. (2)36.141 (1985), pp. 57–69. 30

1985
[18]

Integers free of large prime factors and the Riemann hypothesis

A. Hildebrand. “Integers free of large prime factors and the Riemann hypothesis”.Mathematika31.2 (1984), pp. 258–271

1984
[19]

M. C. E. Hofmann et al.Moments of random multiplicative functions over function fields.https : //arxiv.org/abs/2408.08309. 2024

2024
[20]

Summation of a random multiplicative function on numbers having few prime factors

B. Hough. “Summation of a random multiplicative function on numbers having few prime factors”. Math. Proc. Cambridge Philos. Soc.150.2 (2011), pp. 193–214

2011
[21]

Asymptotic expansions for the Stirling numbers of the first kind

H.-K. Hwang. “Asymptotic expansions for the Stirling numbers of the first kind”.J. Combin. Theory Ser. A71.2 (1995), pp. 343–351

1995
[22]

Squarefree polynomials and M¨ obius values in short intervals and arithmetic progressions

J. Keating and Z. Rudnick. “Squarefree polynomials and M¨ obius values in short intervals and arithmetic progressions”.Algebra Number Theory10.2 (2016), pp. 375–420

2016
[23]

The variance of the number of prime polynomials in short intervals and in residue classes

J. P. Keating and Z. Rudnick. “The variance of the number of prime polynomials in short intervals and in residue classes”.Int. Math. Res. Not. IMRN2014.1 (2014), pp. 259–288

2014
[24]

On the random Chowla conjecture

O. Klurman, I. D. Shkredov, and M. W. Xu. “On the random Chowla conjecture”.Geom. Funct. Anal. 33.3 (2023), pp. 749–777

2023
[25]

Dependent central limit theorems and invariance principles

D. L. McLeish. “Dependent central limit theorems and invariance principles”.Ann. Probability2 (1974), pp. 620–628

1974
[26]

M´ erai.Smooth polynomials with several prescribed coefficients.https://arxiv.org/abs/2307

L. M´ erai.Smooth polynomials with several prescribed coefficients.https://arxiv.org/abs/2307. 10280. 2024

2024
[27]

A polynomial analogue of the twin prime conjecture

P. Pollack. “A polynomial analogue of the twin prime conjecture”.Proc. Amer. Math. Soc.136.11 (2008), pp. 3775–3784

2008
[28]

Rosen.Number theory in function fields

M. Rosen.Number theory in function fields. Vol. 210. Graduate Texts in Mathematics. Springer-Verlag, New York, 2002, pp. xii+358

2002
[29]

Some problems in analytic number theory for polynomials over a finite field

Z. Rudnick. “Some problems in analytic number theory for polynomials over a finite field”.Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II. Kyung Moon Sa, Seoul, 2014, pp. 443–459

2014
[30]

Square-root cancellation for sums of factorization functions over short intervals in function fields

W. Sawin. “Square-root cancellation for sums of factorization functions over short intervals in function fields”.Duke Math. J.170.5 (2021), pp. 997–1026

2021
[31]

A Brun-Titchmarsh theorem for multiplicative functions

P. Shiu. “A Brun-Titchmarsh theorem for multiplicative functions”.J. Reine Angew. Math.313 (1980), pp. 161–170

1980
[32]

Central limit theorems for random multiplicative functions

K. Soundararajan and M. W. Xu. “Central limit theorems for random multiplicative functions”.J. Anal. Math.151.1 (2023), pp. 343–374

2023
[33]

A model problem for multiplicative chaos in number theory

K. Soundararajan and A. Zaman. “A model problem for multiplicative chaos in number theory”. Enseign. Math.68.3-4 (2022), pp. 307–340

2022
[34]

The fourth moment of DirichletL-functions for the rational function field

N. Tamam. “The fourth moment of DirichletL-functions for the rational function field”.Int. J. Number Theory10.1 (2014), pp. 183–218

2014
[35]

Irregularities in the distributions of primes in function fields

F. Thorne. “Irregularities in the distributions of primes in function fields”.J. Number Theory128.6 (2008), pp. 1784–1794

2008
[36]

Sieve methods for polynomial rings over finite fields

W. A. Webb. “Sieve methods for polynomial rings over finite fields”.J. Number Theory16.3 (1983), pp. 343–355

1983
[37]

Random factorizations and Riemann’s hypothesis

A. Wintner. “Random factorizations and Riemann’s hypothesis”.Duke Math. J.11 (1944), pp. 267– 275

1944
[38]

Better than square-root cancellation for random multiplicative functions

M. W. Xu. “Better than square-root cancellation for random multiplicative functions”.Trans. Amer. Math. Soc. Ser. B11 (2024), pp. 482–507. Department of Mathematics, University of California, Berkeley, Berkeley, CA, 94720, USA Email address:declan.hoban@berkeley.edu Department...

2024

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [2]

Sums of random multiplicative functions over function fields with few irreducible factors

D. Aggarwal et al. “Sums of random multiplicative functions over function fields with few irreducible factors”.Math. Proc. Cambridge Philos. Soc.173.3 (2022), pp. 715–726

2022

[2] [1]

The function field Sathe-Selberg formula in arithmetic progressions and ‘short intervals’

A. Afshar and S. Porritt. “The function field Sathe-Selberg formula in arithmetic progressions and ‘short intervals’”.Acta Arith.187.2 (2019), pp. 101–124

2019

[3] [3]

Atherfold and J

C. Atherfold and J. Najnudel.The Fourier coefficients of the critical holomorphic multiplicative chaos. https://arxiv.org/abs/2508.13849. 2025

arXiv 2025

[4] [4]

Prime polynomials in short intervals and in arithmetic progressions

E. Bank, L. Bary-Soroker, and L. Rosenzweig. “Prime polynomials in short intervals and in arithmetic progressions”.Duke Math. J.164.2 (2015), pp. 277–295

2015

[5] [5]

Bary-Soroker and N

L. Bary-Soroker and N. Goldgraber.Product Sets of Arithmetic Progressions in Function Fields.https: //arxiv.org/abs/2309.05046. 2023

arXiv 2023

[6] [6]

Random multiplicative functions in short intervals

S. Chatterjee and K. Soundararajan. “Random multiplicative functions in short intervals”.Int. Math. Res. Not. IMRN2012.3 (2012), pp. 479–492

2012

[7] [7]

Uniform estimates for almost primes over finite fields

D. Elboim and O. Gorodetsky. “Uniform estimates for almost primes over finite fields”.Proc. Amer. Math. Soc.150.7 (2022), pp. 2807–2822

2022

[8] [8]

Gorodetsky and M

O. Gorodetsky and M. D. Wong.Multiplicative chaos measure for multiplicative functions: theL 1- regime.https://arxiv.org/abs/2503.10555. 2025

arXiv 2025