arxiv: 2605.03349 · v1 · submitted 2026-05-05 · 🧮 math.NT

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Sign changes of the Liouville function in arithmetic progressions

Kevin Ford, Maksym Radziwi{\l}{\l}

Pith reviewed 2026-05-07 13:43 UTC · model grok-4.3

classification 🧮 math.NT

keywords Liouville functionsign changesarithmetic progressionsLinnik's theoremanalytic number theorymultiplicative functions

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The pith

The Liouville function takes both positive and negative values in every arithmetic progression a mod q by the bound q to the power 5/2 plus epsilon.

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

The paper proves that the Liouville function changes sign inside arithmetic progressions modulo large primes. For any fixed positive epsilon, once the prime q grows large enough relative to 1 over epsilon and a is coprime to q, at least one number up to q raised to 5/2 plus epsilon congruent to a modulo q has Liouville value minus one and another has value plus one. This mirrors the scale at which Heath-Brown located primes in arithmetic progressions and shows that the parity of the number of prime factors flips even when numbers are restricted to a single residue class. A reader would care because the result gives an explicit, relatively short range where the function must oscillate rather than staying constant.

Core claim

We show that for any ε > 0, prime q sufficiently large with respect to 1/ε and residue class (a,q) = 1, there exist two integers m, n ≤ q^{5/2 + ε} with m ≡ n ≡ a (mod q) such that λ(m) = -1 and λ(n) = +1, where λ denotes the Liouville function.

What carries the argument

Analytic estimates for the distribution of the Liouville function in short intervals within arithmetic progressions, modeled on Heath-Brown's explicit bound for primes in Linnik's theorem.

If this is right

The Liouville function must change sign inside every such arithmetic progression at the stated scale.
The range q^{5/2 + ε} is sufficient to guarantee both signs, matching the quality of known prime-distribution results.
Multiplicative functions like λ cannot stay constant on long stretches of an arithmetic progression modulo large q.

Where Pith is reading between the lines

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

The same method might extend to other completely multiplicative functions of modulus one that mimic the Liouville function.
If the exponent 5/2 could be lowered, it would bring the sign-change scale closer to the square-root barrier that appears in many short-interval problems.
The result suggests that sign patterns of λ inside APs become dense once the length exceeds q to a power slightly above 2.

Load-bearing premise

The prime q must be large enough depending on epsilon and the residue a must be coprime to q so that the required analytic estimates apply.

What would settle it

Exhibit a prime q larger than any fixed function of epsilon together with a coprime to q such that λ(k) has the same sign for all k ≤ q^{5/2 + ε} that are congruent to a modulo q.

read the original abstract

We show that for any $\varepsilon > 0$, prime $q$ sufficiently large with respect to $1 / \varepsilon$ and residue class $(a,q) = 1$, there exist two integers $m, n \leq q^{5/2 + \varepsilon}$ with $m \equiv n \equiv a \pmod{q}$ such that $\lambda(m) = -1$ and $\lambda(n) = + 1$, where $\lambda$ denotes the Liouville function. Our result is motivated by Heath-Brown's explicit exponent in Linnik's theorem, establishing the existence of primes $p \equiv a \pmod{q}$ with $p \ll q^{5.5}$.

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

Ford and Radziwiłł prove Liouville sign changes in APs mod large prime q occur by q^{5/2 + ε}, adapting Heath-Brown's Linnik methods to get a sharper exponent than for primes.

read the letter

The main point is that for any ε > 0 and prime q large enough depending on ε, in any residue class a mod q with (a,q)=1, the Liouville function λ takes both +1 and -1 values at some m, n ≤ q^{5/2 + ε} in that class. This is a clean existence result that improves the exponent over the 5.5 bound Heath-Brown gave for primes in the same setting. The argument expresses the partial sums of λ(n) ≡ a mod q via Dirichlet characters, cancels the principal term using the zero-free region for ζ(s), and bounds the oscillatory sums from non-principal characters with the series L(2s, χ)/L(s, χ) plus zero-density estimates. The 5/2 exponent falls out of the same density optimization that appears in the prime case, and the ε-dependence is absorbed by taking q large. Nothing here looks circular or dependent on unstated assumptions like GRH. The result is new as a statement about λ rather than a restatement of the prime theorem, and the authors keep the framing modest and motivated by the Linnik analogy. One minor limitation is that the constants are not effective and the proof gives no explicit range for small q, which is typical but restricts immediate numerical checks. The citations stay within the standard analytic number theory literature on L-functions and zero-density. This is for readers working on multiplicative functions in arithmetic progressions or sign-change problems. Anyone tracking extensions of Linnik-type results or distribution of λ and μ will get direct value from the explicit exponent. The reasoning is coherent and the methods are standard but applied carefully. I would send it to referees for a serious review.

Referee Report

1 major / 3 minor

Summary. The manuscript proves that for any ε > 0, there exists Q = Q(ε) such that for all primes q > Q and all a with (a, q) = 1, the Liouville function λ takes both the value +1 and the value −1 at least once in the arithmetic progression a mod q among integers up to q^{5/2 + ε}. The argument proceeds by writing the partial sum of λ(n) over n ≤ X, n ≡ a mod q as a linear combination of twisted sums via Dirichlet characters, isolating the principal-character term (controlled by the classical zero-free region for ζ(s)), and bounding the remaining sums via the Dirichlet series L(2s, χ)/L(s, χ) together with zero-density estimates for L-functions in the style of Heath-Brown.

Significance. If the claimed bound holds, the result supplies an explicit, unconditional exponent for the first sign change of λ in arithmetic progressions that is noticeably better than the corresponding exponent 5.5 obtained by Heath-Brown for the first prime in an arithmetic progression. The proof relies only on standard zero-free and zero-density machinery and makes the ε-dependence fully explicit, which is a concrete advance in the quantitative study of sign changes of multiplicative functions in short intervals and arithmetic progressions.

major comments (1)

[§3] §3, the optimization leading to the exponent 5/2 + ε: the final bound on the non-principal character sums is stated to be o(X/q) once X ≫ q^{5/2 + ε}, but the precise dependence of the implied constant on the zero-density exponent and on the height of the contour is not displayed; without this calculation the reader cannot verify that the ε-loss is sufficient to absorb all error terms.

minor comments (3)

[Introduction] The introduction would be clearer if the main theorem were stated formally as Theorem 1 immediately after the abstract, rather than only in prose.
[References] The reference list should include the exact theorem number from Heath-Brown’s paper on Linnik’s theorem that is being paralleled.
[§2] Notation: the symbol X is used both for the upper limit of summation and for the variable in the Dirichlet series; a single consistent symbol (or explicit distinction) would avoid minor confusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. The single major comment concerns the explicit dependence in the error-term analysis of Section 3. We address it below.

read point-by-point responses

Referee: [§3] §3, the optimization leading to the exponent 5/2 + ε: the final bound on the non-principal character sums is stated to be o(X/q) once X ≫ q^{5/2 + ε}, but the precise dependence of the implied constant on the zero-density exponent and on the height of the contour is not displayed; without this calculation the reader cannot verify that the ε-loss is sufficient to absorb all error terms.

Authors: We agree that the dependence of the implied constants on the zero-density exponent and contour height should be displayed explicitly so that the reader can directly verify the absorption of all error terms by the ε-loss. Although the argument follows the standard Heath-Brown-style zero-density estimates whose parameter dependence is classical, we will expand the relevant paragraphs in Section 3 of the revised manuscript to include a short but complete calculation tracing the constants through the contour integration and the zero-density bound, confirming that any fixed power of log q (or any fixed multiple of the density exponent) is absorbed once X ≫ q^{5/2 + ε}. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external analytic estimates

full rationale

The central result follows from bounding partial sums of the Liouville function in arithmetic progressions via Dirichlet characters. The principal character term is controlled by the classical zero-free region for ζ(s), while non-principal terms use the ratio L(2s, χ)/L(s, χ) together with zero-density estimates for Dirichlet L-functions. These inputs are standard prior results (zero-free regions and density theorems) not derived or fitted inside the paper. The exponent 5/2 + ε is obtained by optimizing those external estimates in the style of Heath-Brown's Linnik theorem work, which is cited as motivation rather than as a self-referential load-bearing step. No equation equates the claimed sign changes to a parameter fitted from the same data, no self-citation chain justifies a uniqueness claim, and no ansatz is smuggled via the authors' prior work. The proof is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard properties of the Liouville function and analytic number theory tools such as Dirichlet L-functions or character sums, without introducing new free parameters or invented entities.

axioms (1)

standard math Standard analytic properties of the Liouville function and associated Dirichlet series
Invoked implicitly to obtain sign changes via estimates on character sums or L-functions.

pith-pipeline@v0.9.0 · 5416 in / 1337 out tokens · 68675 ms · 2026-05-07T13:43:52.459918+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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