On variants of Chowla's conjecture

Krishnarjun Krishnamoorthy

REVIEW 3 major objections 4 minor 22 references

On variants of Chowla's conjecture

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For small prime sets, shifted correlation averages of ±1-valued completely multiplicative functions equal an Euler product of local factors, and the attainable values form a complete interval.

desk verdict Nice elementary proofs of two known Chowla-variant theorems, plus a new spectrum result whose proof has a fixable gap in the negative-α_H case. read the letter →

arxiv 2501.10962 v2 pith:Q5HOM2I4 submitted 2025-01-19 math.NT math.CO

Krishnarjun Krishnamoorthy This is my paper · ORCID

classification math.NTmath.CO MSC 11N3711P3211N3511T06

keywords Chowla'sconjectureLiouvillefunctioncompletelymultiplicativefunctionsshiftedconvolutionsumsspectrumsmallsetsofprimesnaturaldensitytwo-pointcorrelations

topics Goldbach's 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

This paper studies averages of shifted products $\lambda_P(n+h_1)\cdots\lambda_P(n+h_d)$, where $\lambda_P$ is the completely multiplicative function equal to $-1$ at primes in $P$ and $1$ elsewhere. It proves that when $P$ is a small set of primes, the long-run average has an exact Euler-product formula whose factors are local densities, one per prime. That makes the global correlation a product of independent local contributions even though the shifted product is not multiplicative. It also proves that only the constant function can achieve a correlation of magnitude $1$, and that the set of all possible correlation values for a fixed shift set is a full interval union. These are variants of Chowla's conjecture, which predicts such averages vanish for the Liouville function.

What carries the argument

The carrying object is the set $N_H^P=\{n:\Lambda_H^P(n)=-1\}$, whose natural density $\eta_H^P$ is the quantity being computed. Lemma 2 gives $N_{H_1\triangle H_2}^{P_1\triangle P_2}=N_{H_1}^{P_1}\triangle N_{H_1}^{P_2}\triangle N_{H_2}^{P_1}\triangle N_{H_2}^{P_2}$, so primes and shifts can be added one at a time. A double induction approximates $N_H^P$ from inside and outside by finite unions of arithmetic progressions with moduli made only from primes in $P$, yielding the density recursion $\eta_H^P=\eta_H^{P'}(1-\eta_p^H)+\eta_p^H(1-\eta_H^{P'})$, which factors into $1-2\eta_H^P=\prod_{p\in P}(1-2\eta_p^H)$. Inequality (3.9) then controls the passage from finite to small infinite $P$ by the tail sum over primes outside a finite subset. For Theorem 2, the machinery is the symmetric-difference closure of the family of shift sets with $|\kappa_H^P|=1$, combined with a cited two-point correlation bound that rules out values of magnitude $1$.

What would settle it

Compute $S(x)=(1/x)\sum_{n\le x}\lambda_{\{2\}}(n)\lambda_{\{2\}}(n+4)\lambda_{\{2\}}(n+6)$, where $\lambda_{\{2\}}(p)=-1$ only at $p=2$ and is $1$ at all other primes. The paper's formula predicts the limit is $1-2\cdot(1/6)=2/3$; a persistent deviation as $x$ grows would refute Theorem 1. Alternatively, exhibiting a value in $(0,1)$ that no small set $P$ realizes would refute Proposition 3.

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Extended reading notes

Core claim

The central claim is Theorem 1: for any finite shift set $H$ and any small prime set $P$, the limit $\kappa_H^P=\lim_{x\to\infty}\frac{1}{x}\sum_{n\le x}\prod_{h\in H}\lambda_P(n+h)$ exists and equals $\prod_{p\in P}(1-2\eta_p^H)$, where $\eta_p^H=d/(p+1)$ whenever $p$ divides none of the differences $h_i-h_j$, and exceptional primes have explicitly computable constants. The proof works through the density of the set $N_H^P=\{n:\Lambda_H^P(n)=-1\}$, which is shown to be approximable by finite unions of arithmetic progressions whose moduli use only primes from $P$. Theorem 2 then says that if some non-empty $H$ has $|\kappa_H^P|=1$, then $P$ is empty, so non-constant functions never produce perfect correlation; the proof reduces this to the two-point case via a symmetric-difference closure property. Finally, the spectrum $\Gamma_H$, the closure of all values $\kappa_H^P$ over small $P$, is exactly $[\alpha_H,1]\cup[0,1]$.

Load-bearing premise

Lemma 6 assumes that for any interval $(a,b)$ inside $(0,1)$ one can find finitely many new large primes whose per-prime factors multiply into $(a,b)$, and the only justification given is the divergence of the sum of reciprocals of primes; if this density statement failed, the claimed spectrum could omit attainable values.

Editorial extensions

If this is right

For any small prime set $P$ and finite shift set $H$, the average $\kappa_H^P$ exists and equals $\prod_{p\in P}(1-2\eta_p^H)$, so the average is fixed exactly by per-prime data.
For every non-exceptional prime, $\eta_p^H=d/(p+1)$; therefore the convergence of the product is the same as $P$ being small, and the product vanishes exactly when the sum of reciprocals diverges.
No non-constant $\pm1$-valued completely multiplicative function can have a finite-shift correlation of magnitude $1$; every such average satisfies $\liminf<1$ and $\limsup>-1$.
For a fixed $H$, the closure of all values $\kappa_H^P$ over small $P$ is $[\alpha_H,1]\cup[0,1]$, so every value in that range is realized by some small set of primes.
The recursive procedure also computes exceptional factors explicitly, for example $\eta^{\{0,4,6\}}_2=1/6$ and $\eta^{\{0,4,6\}}_3=5/12$.

Reading between the lines

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

The same one-prime-at-a-time recursion should carry over to correlations of $\pm1$-valued functions twisted by Dirichlet characters, with $\eta_p^H$ replaced by counts of roots modulo prime powers; the paper does not pursue this extension.
Lemma 6's use of the divergence of reciprocals suggests a stronger quantitative statement: the finite-product approximants can be chosen so that the error at stage $i$ decays like a tail of the prime harmonic series, which would give explicit convergence rates for the spectrum construction.
Theorem 2 would become fully elementary if the two-point correlation bound it invokes could be replaced by a direct combinatorial argument; Remark 2 already points out that the limsup/liminf version would require a stronger form of Lemma 4, so the dependence is genuine.

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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

First thing to know: this paper re-proves two known results, Klurman's Theorem 1 and Teräväinen's Theorem 2, with a mostly elementary combinatorial method, and adds a new spectrum result (Proposition 3). The elementary machinery is the real value: the symmetric-difference algebra and the density approximation argument are clean and worth having. The spectrum statement is natural and probably true, but the proof as written has a genuine gap in the negative-α_H case.

The good parts: Theorem 1's proof is self-contained and sound. The limiting argument for small sets via the tail bound (3.9) works. Theorem 2's reduction to the Matomäki-Radziwill theorem is correct, and the lemma borrowed from Wildon is a neat way to get a two-point set. The paper is honest about what is new: both theorems are explicitly attributed to earlier work, and Proposition 3 is the only claimed new result.

The soft spots, in proportion. In Section 5, proof of Proposition 3, the α_H < 0 case: the text sets α := β/α_H (which is in (0,1)) and then chooses P with κ_P^H = 1 − 2α. That does not follow from Lemma 6, which only provides positive κ values, and 1 − 2α is negative whenever β < α_H/2. Also, for primes larger than X, the factors 1 − 2η_p^H are positive, so κ_P^H cannot be negative at all. The fix is immediate: choose P with κ_P^H = β/α_H (which does lie in (0,1)) and then multiply by the α_H factor from p. As written, the proof does not cover part of the interval, so this is load-bearing for the new result, but it is a small repair, not a deep flaw.

Lemma 6 is also terse: the assertion that you can hit any target in (0,1) using disjoint finite sets of large primes because the reciprocal-prime sum diverges needs more than one sentence. The construction is plausible and standard, but the write-up should expand it.

The reverse inclusion (κ_P ≥ α_H) is fine, and there is no circularity or citation weirdness.

Who this is for: analytic number theorists working on Chowla-type correlations. It deserves a serious referee; it should not be desk rejected. The referee should require the Proposition 3 correction and an expanded Lemma 6. After that, it's a solid short paper.

Referee Report

3 major / 4 minor

Summary. The paper studies shifted convolution sums of completely multiplicative functions λ_P taking values in {±1}, where P is a set of primes. For a finite shift set H, it defines κH_P as the limiting average of ΛH_P(n) = λ_P(n+h1)...λ_P(n+hd). Theorem 1 asserts that for small P (convergent sum of reciprocals), κH_P equals an Euler product ∏_{p∈P}(1-2ηH_p), with ηH_p = d/(p+1) for non-exceptional primes. Theorem 2 states that no non-constant λ_P has a correlation with absolute value 1, following from a result of Matomäki and Radziwill. Proposition 3 characterizes the spectrum Γ_H, the closure of all attainable κH_P values, as [α_H,1] ∪ [0,1], where α_H = inf_p(1-2ηH_p). The proofs use a symmetric-difference algebra for the sets N_H^P where ΛH_P = -1, and a density approximation argument.

Significance. The paper gives elementary, self-contained proofs of known correlation formulas for λ_P and obtains a complete description of the spectrum of possible correlations for small prime sets. The symmetric-difference viewpoint is elegant and provides a concrete way to compute ηH_p (as in Example 1). If the spectrum theorem is correct, it is a new structural result for this family of multiplicative functions. The main theorems are stated clearly, and the dependence on external deep input (Matomäki-Radziwill) is explicit and limited, which is a strength. No code or machine-checked proofs are provided, but the arguments are conceptually transparent and likely repairable.

major comments (3)

[Section 5, proof of Proposition 3 (α_H < 0 case)] The construction of P ∪ {p} is algebraically wrong. With α := β/α_H, choosing P such that κH_P = 1-2α gives κH_{P∪{p}} = (1-2α)α_H = α_H - 2β, which equals β only in the special case α_H = 3β. Moreover, if α > 1/2, then 1-2α is negative, impossible for primes p > X (where the factors are positive). The correct choice is a small set P with κH_P = β/α_H, which lies in (0,1) and is attainable by Lemma 6; then κH_{P∪{p}} = (β/α_H)·α_H = β. This error is load-bearing for the inclusion [α_H,1] ⊆ Γ_H and must be fixed.
[Section 3, Proposition 2 (induction structure)] The double induction is not fully specified. When the projection modulo p is constant, the proof replaces H by H1 = (H - i1)/p, which has max H1 < max H but the same cardinality as H. The later statement 'we may then apply induction on |H|' is unjustified, since |H_r| = |H| at the terminal stage. A well-founded induction on a measure such as (|H|, max H) would repair the argument, but as written the induction is incomplete and this is used to establish Theorem 1.
[Section 5, Lemma 6] The assertion 'Such a choice is possible because the sum of reciprocals of primes is divergent' is not proved, yet it is the core of the lemma's construction. It is true (the finite subset products of (1-2d/(p+1)) are dense in (0,1) because ∑ 2d/(p+1) diverges and the summands tend to 0), but the one-sentence justification is too terse for a load-bearing step in Proposition 3. A short proof or reference would make the lemma self-contained.

minor comments (4)

[Abstract and Introduction] 'Combinatorical' should be 'combinatorial'.
[Section 3, equation (3.9)] The bound δ+(N_H^{P\P_i}) ≤ ∑_{p∈P\P_i} ηH_p follows from Lemma 2 and (3.5), but this is not stated; please add a sentence making the use of Lemma 2 explicit.
[Section 5, proof of Proposition 3, final paragraph] The chain 'κH_P ≥ κH_{P\{p}}α_H ≥ α_H' is misleading: the first inequality requires a case split on the sign of κH_{P\{p}}, and the second uses κH_{P\{p}} ≤ 1. The conclusion is true because every factor lies in [α_H,1] and products of such factors are ≥ α_H, but the written chain should be clarified.
[Notation] The empty set is denoted φ, which may be confused with the totient function; consider using ∅.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorems are proved from external black boxes (Matomäki–Radziwill, Wildon's lemma) and internally proved elementary lemmas; the spectrum argument does not assume its conclusion.

full rationale

I walked the derivation chain of Theorems 1 and 2 and Proposition 3. Theorem 1 is proved from Proposition 2 and a limiting argument for small sets; it is explicitly said to be deducible from Klurman [10], but the paper supplies an independent combinatorial proof, so the citation is not load-bearing. Theorem 2 is introduced as following from Teravainen [20], but the paper's proof instead uses Matomäki–Radziwill (Theorem 4) and Wildon's lemma (Lemma 5) to reduce to the two-point case; these are external inputs and do not presuppose the target statement. Proposition 3 uses Lemma 6, whose proof asserts finite choices of primes with prescribed products based on divergence of the reciprocal-prime series; this is terse and perhaps underjustified, and the α_H<0 branch of Proposition 3 appears to contain an algebraic slip (the choice α := β/α_H does not make (1−2α)α_H equal β), but such a mistake is a correctness issue, not a circularity. The paper contains no self-citation chain, no uniqueness theorem imported from the authors' prior work, no fitted parameter renamed as a prediction, and no definition of the spectrum that assumes the interval description. Hence the honest finding is no significant circularity.

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

The paper introduces no new free parameters: the constants eta_p^H are computed exactly from the combinatorics of H modulo p, not fitted to data. It relies on standard analytic number theory results (Matomäki-Radziwill, Euler, CRT) and standard density arguments. No new particles, forces, or dimensions are postulated.

assumptions (4)

domain assumption Matomäki-Radziwill theorem (Theorem 4 in the paper): for any non-empty P and h>=1, the two-point correlation average is bounded away from 1 for large x.
Used as the main arithmetic input to prove Theorem 2. The paper explicitly calls it "the main arithmetic input" in the overview and Remark 1. It is a deep external result, not proved in this paper.
standard math Divergence of the sum of reciprocals of primes (Euler).
Used in the proof of Lemma 6 to construct finite sets with prescribed product values, and in section 3.2 to show the tail sum tends to 0 for small sets.
standard math Chinese remainder theorem.
Used in Lemma 3 and Proposition 1 to multiply densities of arithmetic progressions with coprime moduli.
standard math Basic properties of natural density for finite unions of arithmetic progressions.
Used in Lemma 3 to establish existence of densities in the algebra A_P and to justify the independence formula in Proposition 1.

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Cite this review

Pith. "Pith review of On variants of Chowla's conjecture." pith.science (2026). https://pith.science/paper/Q5HOM2I4

@misc{pith2026250110962,
  author       = {Pith},
  title        = {Pith review of: On variants of Chowla's conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5HOM2I4}},
  note         = {Machine review of arXiv:2501.10962}
}

abstract

We study the shifted convolution sums associated to completely multiplicative functions taking values in $\{\pm 1\}$ and give combinatorical proofs of two recent results in the direction of Chowla's conjecture. We also determine the corresponding "spectrum".

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Peter Borwein, Stephen K. K. Choi, and Michael Coons. Complete ly multiplicative functions taking values in {−1,1}. Trans. Amer. Math. Soc. , 362(12):6279–6291, 2010

work page 2010
[2]

Siegel zeros and sarnak’s conjecture, 2021

Jake Chinis. Siegel zeros and sarnak’s conjecture, 2021

work page 2021
[3]

S. Chowla. The Riemann hypothesis and Hilbert’s tenth problem , volume Vol. 4 of Mathematics and its Applications. Gordon and Breach Science Publishers, New York-London-Paris, 1965

work page 1965
[4]

On the convolution of the Liouville function under the existence of Siegel zeros

Jean-Marie De Koninck, L´ aszl´ o Germ´ an, and Imre K´ atai. On the convolution of the Liouville function under the existence of Siegel zeros. Lith. Math. J. , 55(3):331–342, 2015

work page 2015
[5]

Germ´ an and I

L. Germ´ an and I. K´ atai. On multiplicative functions on consecutive integers. Lith. Math. J., 50(1):43– 53, 2010

work page 2010
[6]

Soundararajan

Andrew Granville and K. Soundararajan. The spectrum of multiplic ative functions. Ann. of Math. (2), 153(2):407–470, 2001

work page 2001
[7]

Expansion, divisibility and parity, 2021

Harald Andr´ es Helfgott and Maksym Radziwi/suppress l/suppress l. Expansion, divisibility and parity, 2021

work page 2021
[8]

Collection closed under sym- metric diﬀerence and translation

Mark Wildon (https://mathoverﬂow.net/users/7709/mark wildo n). Collection closed under sym- metric diﬀerence and translation. MathOverﬂow. URL:https://mathoverﬂow.net/q/484822 (version: 2024-12-26)

work page 2024

Show all 22 references

[9]

The chowla conjecture and landau-siegel zeroes, 2024

Mikko Jaskari and Stelios Sachpazis. The chowla conjecture and landau-siegel zeroes, 2024. 14 KRISHNARJUN KRISHNAMOORTHY

2024
[10]

Correlations of multiplicative functions and applic ations

Oleksiy Klurman. Correlations of multiplicative functions and applic ations. Compos. Math. , 153(8):1622–1657, 2017

2017
[11]

Multiplicative functionsin short intervals

Kaisa Matom¨ aki and Maksym Radziwi/suppress l /suppress l. Multiplicative functionsin short intervals. Ann. of Math. (2), 183(3):1015–1056, 2016

2016
[12]

An averaged form of Chowla’s conjecture

Kaisa Matom¨ aki, Maksym Radziwi/suppress l/suppress l, and Terence Tao. An averaged form of Chowla’s conjecture. Algebra Number Theory, 9(9):2167–2196, 2015

2015
[13]

Higher uniformity of bounded multiplicative functions in short intervals on av erage

Kaisa Matom¨ aki, Maksym Radziwi/suppress l/suppress l, Terence Tao, Joni Ter¨ av¨ ainen, and Tamar Ziegler. Higher uniformity of bounded multiplicative functions in short intervals on av erage. Ann. of Math. (2) , 197(2):739–857, 2023

2023
[14]

Improved bounds for the two-point logarithmicchowla conjecture, 2023

C´ edric Pilatte. Improved bounds for the two-point logarithmicchowla conjecture, 2023

2023
[15]

Chowla’s conjecture: from the Liouville functionto the Moebius function

Olivier Ramar´ e. Chowla’s conjecture: from the Liouville functionto the Moebius function. In Ergodic theory and dynamical systems in their interactions with ari thmetics and combinatorics , volume 2213 of Lecture Notes in Math. , pages 317–323. Springer, Cham, 2018

2018
[16]

The logarithmically averaged Chowla and Elliott conjectures for two-point correlations

Terence Tao. The logarithmically averaged Chowla and Elliott conjectures for two-point correlations. Forum Math. Pi , 4:e8, 36, 2016

2016
[17]

Odd order cases of the logarithmically averaged Chowla conjecture

Terence Tao and Joni Ter¨ av¨ ainen. Odd order cases of the logarithmically averaged Chowla conjecture. J. Th´ eor. Nombres Bordeaux, 30(3):997–1015, 2018

2018
[18]

The structure of logarithmically averaged correlations of mul- tiplicative functions, with applications to the Chowla and Elliott conjec tures

Terence Tao and Joni Ter¨ av¨ ainen. The structure of logarithmically averaged correlations of mul- tiplicative functions, with applications to the Chowla and Elliott conjec tures. Duke Math. J. , 168(11):1977–2027, 2019

1977
[19]

The Hardy-Littlewood-Chowla conjecture in the presence of a Siegel zero

Terence Tao and Joni Ter¨ av¨ ainen. The Hardy-Littlewood-Chowla conjecture in the presence of a Siegel zero. J. Lond. Math. Soc. (2) , 106(4):3317–3378, 2022

2022
[20]

On the Liouville function at polynomial arguments

Joni Ter¨ av¨ ainen. On the Liouville function at polynomial arguments. Amer. J. Math. , 146(4):1115– 1167, 2024

2024
[21]

The Theory of Measure in Arithmetical Semi-Groups

Aurel Wintner. The Theory of Measure in Arithmetical Semi-Groups . publisher unknown, Baltimore, MD, 1944

1944
[22]

E. Wirsing. Das asymptotische Verhalten von Summen ¨ uber mult iplikative Funktionen. II. Acta Math. Acad. Sci. Hungar. , 18:411–467, 1967. Email address, Krishnarjun Krishnamoorthy: krishnarjunmaths@outlook.com Beijing Institute of Mathemtical Sciences and Application s (BIMS...

1967

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Peter Borwein, Stephen K. K. Choi, and Michael Coons. Complete ly multiplicative functions taking values in {−1,1}. Trans. Amer. Math. Soc. , 362(12):6279–6291, 2010

work page 2010

[2] [2]

Siegel zeros and sarnak’s conjecture, 2021

Jake Chinis. Siegel zeros and sarnak’s conjecture, 2021

work page 2021

[3] [3]

S. Chowla. The Riemann hypothesis and Hilbert’s tenth problem , volume Vol. 4 of Mathematics and its Applications. Gordon and Breach Science Publishers, New York-London-Paris, 1965

work page 1965

[4] [4]

On the convolution of the Liouville function under the existence of Siegel zeros

Jean-Marie De Koninck, L´ aszl´ o Germ´ an, and Imre K´ atai. On the convolution of the Liouville function under the existence of Siegel zeros. Lith. Math. J. , 55(3):331–342, 2015

work page 2015

[5] [5]

Germ´ an and I

L. Germ´ an and I. K´ atai. On multiplicative functions on consecutive integers. Lith. Math. J., 50(1):43– 53, 2010

work page 2010

[6] [6]

Soundararajan

Andrew Granville and K. Soundararajan. The spectrum of multiplic ative functions. Ann. of Math. (2), 153(2):407–470, 2001

work page 2001

[7] [7]

Expansion, divisibility and parity, 2021

Harald Andr´ es Helfgott and Maksym Radziwi/suppress l/suppress l. Expansion, divisibility and parity, 2021

work page 2021

[8] [8]

Collection closed under sym- metric diﬀerence and translation

Mark Wildon (https://mathoverﬂow.net/users/7709/mark wildo n). Collection closed under sym- metric diﬀerence and translation. MathOverﬂow. URL:https://mathoverﬂow.net/q/484822 (version: 2024-12-26)

work page 2024