Pith. sign in

REVIEW 2 minor 22 references

Linnik's problem for multiplicative functions

T0 review · 0 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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

Pith's one-line read For multiplicative functions, sign changes occur in every residue class a mod q at scale q^{2+o(1)}, unless the sign strongly pretends to be a real Dirichlet character mod q.

desk verdict Matomäki-Teräväinen push the multiplicative Linnik problem to the square-root barrier for general h, with only the real-character pretense case left as obstruction. read the letter →

arxiv 2605.27833 v1 pith:7KZZ7R4L submitted 2026-05-27 math.NT

Kaisa Matomäki , Joni Teräväinen This is my paper · ORCID
classification math.NT
keywords multiplicativefunctionsLinnik'sproblemsignchangesarithmeticprogressionsDirichletcharactersLiouvillefunctionsquare-freeintegerspretentiousdistance
topics The Riemann Hypothesis
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 studies an analogue of Linnik's problem for multiplicative functions h from naturals to nonzero reals. It asks how large one must search to guarantee square-free n1 and n2 congruent to a given a mod q with h taking both negative and positive values. The result establishes that n1 and n2 exist at most q to the power 2 plus little-o(1) except precisely when the sign of h strongly pretends to be a real Dirichlet character modulo q. This places sign changes at essentially the square-root barrier for all but this natural obstruction case, and for the Liouville function it removes the previous gap to the GRH-conditional bound up to q to the o(1) factors.

What carries the argument

The technical notion that the sign of h 'strongly pretends' to be a real Dirichlet character modulo q, which is isolated as the sole obstruction to the q^{2+o(1)} bound.

What would settle it

A concrete multiplicative function h together with modulus q and residue a where the smallest pair of square-free n1,n2 ≡ a mod q with opposite signs for h exceeds q^{2+ε} for some fixed ε>0, yet the sign of h does not strongly pretend to be any real character mod q.

Watch

Extended reading notes

Core claim

We show that one can always find such integers with n1,n2≤q^{2+o(1)}, unless the sign of h strongly pretends to be a real Dirichlet character modulo q. Thus, apart from this natural character obstruction, sign changes of a multiplicative function occur in every reduced residue class at a scale corresponding essentially to the square root barrier. In the special case of the Liouville function λ this improves on a recent result of Ford and Radziwiłł and matches, up to q^{o(1)} factors, what was previously known conditionally under the generalized Riemann hypothesis.

Load-bearing premise

The only case where the q^{2+o(1)} bound can fail is when the sign of h strongly pretends to be a real Dirichlet character modulo q, under the standard definition and properties of multiplicative functions.

Editorial extensions

If this is right

  • Square-free integers n1 and n2 ≡ a mod q with h(n1)<0<h(n2) exist below q^{2+o(1)} for any reduced a, except in the strong pretense case.
  • The result applies uniformly to every multiplicative function h taking nonzero real values.
  • For the Liouville function the bound n1,n2 ≤ q^{2+o(1)} is unconditional and essentially matches the GRH-conditional scale.
  • Sign changes are guaranteed inside every arithmetic progression at a scale no worse than the square-root barrier once the character obstruction is removed.

Reading between the lines

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

  • The result implies that pretentiousness to real characters is the dominant barrier for sign distribution questions in arithmetic progressions.
  • One could check whether the same scale holds when h is replaced by products of two or more multiplicative functions.
  • The o(1) term in the exponent might be made explicit with quantitative versions of the pretentious distance estimates used in the argument.
  • The method may adapt to show that sign changes persist even when restricting to numbers with a fixed number of prime factors.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper proves a multiplicative-function analogue of Linnik's problem. For any multiplicative h: N → R eq 0 and reduced residue a mod q, it shows that square-free n1, n2 ≡ a mod q with h(n1) < 0 < h(n2) exist with n1, n2 ≤ q^{2+o(1)}, unless sign(h) strongly pretends to be a real Dirichlet character mod q. The result improves the unconditional bound for the Liouville function λ beyond Ford–Radziwiłł and matches the GRH-conditional square-root barrier up to q^{o(1)} factors, with the character obstruction explicitly isolated as the sole exception.

Significance. If the derivation holds, the result is significant: it establishes that sign changes of multiplicative functions occur in every arithmetic progression at essentially the square-root barrier once the natural character obstruction is removed. The clean, parameter-free statement with an explicit exceptional case, reliance on standard multiplicative-function properties, and the matching of GRH-conditional bounds unconditionally for λ constitute clear advances. The work supplies a falsifiable prediction (the q^{2+o(1)} bound outside the pretense case) that can be checked numerically for small q.

minor comments (2)
  1. The abstract uses the LaTeX fragment “Radziwi{\l}{\l}”; ensure the published version renders the Polish ł correctly in both abstract and bibliography.
  2. The definition of “strongly pretends” is invoked in the main theorem statement; a self-contained recall of this notion (even if standard) in §1 would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper states a direct theorem that square-free sign changes for multiplicative h occur in every reduced residue class a mod q at scale q^{2+o(1)}, except when sign(h) strongly pretends to be a real Dirichlet character mod q. This exception is explicitly identified as the sole obstruction and is defined via a standard technical notion rather than being fitted or self-referential. The derivation is described as relying only on the usual multiplicative-function axioms plus the definition of strong pretense; no equation or step is shown to reduce by construction to a fitted input, a self-citation, or a renamed known result. The improvement over Ford–Radziwiłł is external and the GRH comparison is conditional, leaving the unconditional claim self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim rests on the standard definition of multiplicative functions, the technical notion of 'strongly pretends to be a real Dirichlet character', and background results from analytic number theory; no free parameters or invented entities are visible in the abstract.

assumptions (1)
  • standard math Standard properties of multiplicative functions and the definition of 'strongly pretends to be a real Dirichlet character'
    Invoked to isolate the exceptional case in the main statement.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Linnik's problem for multiplicative functions." pith.science (2026). https://pith.science/paper/7KZZ7R4L

@misc{pith2026260527833,
  author       = {Pith},
  title        = {Pith review of: Linnik's problem for multiplicative functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7KZZ7R4L}},
  note         = {Machine review of arXiv:2605.27833}
}
abstract

We study a multiplicative function analogue of Linnik's problem on the least prime in an arithmetic progression. Let $h\colon \mathbb{N}\to\mathbb{R}\setminus\{0\}$ be a multiplicative function, and let $a \pmod q$ be a reduced residue class. We ask how far one must go before finding square-free integers $n_1,n_2\equiv a \pmod q$ with $h(n_1)<0<h(n_2)$. We show that one can always find such integers with $n_1,n_2\le q^{2+o(1)}$, unless the sign of $h$ strongly pretends to be a real Dirichlet character modulo $q$. Thus, apart from this natural character obstruction, sign changes of a multiplicative function occur in every reduced residue class at a scale corresponding essentially to the square root barrier. In the special case of the Liouville function $\lambda$ this improves on a recent result of Ford and Radziwi{\l}{\l} and matches, up to $q^{o(1)}$ factors, what was previously known conditionally under the generalized Riemann hypothesis.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 1 canonical work pages

  1. [1]

    Balog, A

    A. Balog, A. Granville, and K. Soundararajan. Multiplicative functions in arithmetic progressions. Ann. Math. Qu´ e., 37(1):3–30, 2013

    2013

  2. [2]

    S. Chowla. On the least prime in an arithmetical progression.J. Indian Math. Soc. (N.S.), 1:1–3, 1934

    1934

  3. [3]

    Sign changes of the Liouville function in arithmetic progressions

    Kevin Ford and Maksym Radziwi l l. Sign changes of the liouville function in arithmetic progressions. arXiv e-prints, page arXiv:2605.03349, 2026

    work page Pith review arXiv 2026

  4. [4]

    Friedlander and H

    J. Friedlander and H. Iwaniec.Opera de cribro, volume 57 ofAmerican Mathematical Society Collo- quium Publications. American Mathematical Society, Providence, RI, 2010

    2010

  5. [5]

    Granville, D

    A. Granville, D. Koukoulopoulos, and K. Matom¨ aki. When the sieve works.Duke Math. J., 164(10):1935–1969, 2015

    1935

  6. [6]

    D. J. Grynkiewicz. On extending Pollard’s theorem fort-representable sums.Israel J. Math., 177:413– 439, 2010

    2010

  7. [7]

    D. R. Heath-Brown. Zero-free regions for DirichletL-functions, and the least prime in an arithmetic progression.Proc. London Math. Soc. (3), 1992

    1992

  8. [8]

    Iwaniec and E

    H. Iwaniec and E. Kowalski.Analytic number theory, volume 53 ofAmerican Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004

    2004

Show all 22 references
  1. [9]

    A. Jha. Smallest totient in a residue class.Bull. Lond. Math. Soc., 57(6):1908–1917, 2025

    1908

  2. [10]

    Klurman, A

    O. Klurman, A. P. Mangerel, and J. Ter¨ av¨ ainen. Multiplicative functions in short arithmetic progres- sions.Proc. Lond. Math. Soc. (3), 127(2):366–446, 2023. 48Kaisa Matom ¨aki and Joni Ter ¨av¨ainen

    2023

  3. [11]

    M. A. Korol¨ ev. Kloosterman sums with multiplicative coefficients.Izv. Ross. Akad. Nauk Ser. Mat., 82(4):3–17, 2018

    2018

  4. [12]

    U. V. Linnik. On the least prime in an arithmetic progression. I. The basic theorem.Rec. Math. [Mat. Sbornik] N.S., 15(57):139–178, 1944

    1944

  5. [13]

    U. V. Linnik. On the least prime in an arithmetic progression. II. The Deuring-Heilbronn phenomenon. Rec. Math. [Mat. Sbornik] N.S., 15(57):347–368, 1944

    1944

  6. [14]

    Matom¨ aki and M

    K. Matom¨ aki and M. Radziwi l l. Multiplicative functions in short intervals.Ann. of Math. (2), 183(3):1015–1056, 2016

    2016

  7. [15]

    Matom¨ aki, M

    K. Matom¨ aki, M. Radziwi l l, and T. Tao. An averaged form of Chowla’s conjecture.Algebra Number Theory, 9(9):2167–2196, 2015

    2015

  8. [16]

    Matom¨ aki and X

    K. Matom¨ aki and X. Shao. When the sieve works II.J. Reine Angew. Math., 763:1–24, 2020

    2020

  9. [17]

    Matom¨ aki and J

    K. Matom¨ aki and J. Ter¨ av¨ ainen. Products of primes in arithmetic progressions.J. Reine Angew. Math., 808:193–240, 2024

    2024

  10. [18]

    H. L. Montgomery.Topics in multiplicative number theory, volume Vol. 227 ofLecture Notes in Math- ematics. Springer-Verlag, Berlin-New York, 1971

    1971

  11. [19]

    H. L. Montgomery and R. C. Vaughan.Multiplicative number theory. I. Classical theory, volume 97 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007

    2007

  12. [20]

    Ramar´ e and A

    O. Ramar´ e and A. Walker. Products of primes in arithmetic progressions: a footnote in parity breaking. J. Th´ eor. Nombres Bordeaux, 30(1):219–225, 2018

    2018

  13. [21]

    Tao and V

    T. Tao and V. Vu.Additive combinatorics, volume 105 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006

    2006

  14. [22]

    Xylouris

    T. Xylouris. Linnik’s constant is less than 5.Chebyshevski˘ ı Sb., 19(3(67)):80–94, 2018. Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland Email address:ksmato@utu.fi Department of Pure Mathematics and Mathematical Statistics, University of C...

    2018

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.