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arxiv: 2604.07186 · v1 · submitted 2026-04-08 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

Weighted averages of arithmetic functions and applications to equidistribution

Vitaly Bergelson , Michael Reilly , Florian K. Richter
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:53 UTC · model grok-4.3

classification 🧮 math.NT
keywords weighted averagesuniform distribution mod 1arithmetic functionsHardy fieldsprime factor functionsequidistributionGaussian distribution conditionsquarefree numbers
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The pith

A growth-rate criterion relative to rational polynomials determines exactly when sequences h of arithmetic functions like Ω(n) are uniformly distributed mod 1.

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

The paper first proves a general estimate for weighted averages of the form average of f(θ(n)) whenever the arithmetic function θ obeys a Gaussian distribution condition. It then applies the estimate to obtain a complete if-and-only-if characterization of uniform distribution mod 1 for the sequence h(θ(n)) when h belongs to a Hardy field and grows at most polynomially. A reader cares because the characterization immediately settles equidistribution for all concrete sequences built from prime-factor counts, such as powers of Ω(n), ω(n), or the prime factors of squarefree numbers.

Core claim

If h is a function from a Hardy field with polynomial growth, then (h(θ(n)))_{n∈ℕ} is uniformly distributed mod 1 if and only if one of two mutually exclusive conditions holds: either the distance from h to every rational polynomial grows faster than x log x, or the distance grows faster than sqrt(x) away from every rational polynomial while remaining bounded after division by x for at least one linear polynomial.

What carries the argument

The Gaussian distribution condition on θ(n), which makes the weighted averages of arbitrary f(θ(n)) behave like integrals against a Gaussian density and thereby reduces equidistribution questions to growth-rate comparisons with polynomials.

If this is right

  • The sequence Ω(n)^c is uniformly distributed mod 1 precisely when c is a non-integer greater than 1/2.
  • The same equivalence holds verbatim when Ω(n) is replaced by ω(n) or by Ω(q_n) for q_n the n-th squarefree number.
  • Any h in a Hardy field that stays asymptotically close to a linear polynomial but diverges faster than sqrt(x) from all other rational polynomials yields a uniformly distributed sequence.
  • The weighted-average estimate applies directly to any arithmetic function obeying the Gaussian condition, not only the three examples listed.

Where Pith is reading between the lines

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

  • Numerical checks of partial discrepancies for moderate N could provide independent confirmation of the boundary cases such as c = 1/2 + ε.
  • The same growth-rate test might be adaptable to other additive arithmetic functions whose value distributions are known to be Gaussian-like.
  • The criterion supplies a template for deciding equidistribution of h composed with any θ whose distribution is sufficiently regular, even outside number theory.

Load-bearing premise

The arithmetic function θ(n) satisfies a Gaussian distribution condition that controls its weighted averages.

What would settle it

A concrete computation showing that the Weyl sums for the sequence Ω(n)^{0.6} fail to tend to zero, or that the sums for Ω(n) itself do tend to zero, would falsify the claimed equivalence.

read the original abstract

For a wide range of functions $W\colon\mathbb{N}\to\mathbb{N}$, we establish a general result for estimating weighted averages of the form \[ \mathbb{E}^{W}_{n \le N} f(\vartheta(n))= \frac{1}{W(N)}\sum_{n=1}^N (W(n)-W(n-1))f(\vartheta(n)), \] where $f\colon \{1,\ldots,N\}\to\mathbb{C}$ is an arbitrary function, and $\vartheta(n)$ is any arithmetic function that adheres to a certain Gaussian distribution condition. (In particular, one can take $\vartheta(n)=\Omega(n)$, $\vartheta(n)=\omega(n)$, or $\vartheta(n)=\Omega(q_n)$, where $\Omega(n)$ and $\omega(n)$ count the number of prime factors of $n$ with and without multiplicities respectively, and $q_n$ denotes the $n$-th squarefree number.) As an application of our main theorem, we show that if $h(n)$ is a function from a Hardy field with polynomial growth then $(h(\vartheta(n)))_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if one of the following (mutually exclusive) conditions is satisfied: (i) $\lim_{x\to\infty} \frac{|h(x)-p(x)|}{x \log x}=\infty$ for all $p(x)\in \mathbb{Q}[x]$; (ii) $\lim_{x\to\infty}\frac{|h(x)-p(x)|}{\sqrt{x}}=\infty$ for each $p(x)\in \mathbb{Q}[x]$ and there exists $q(x)\in \mathbb{Q}[x]$ such that $\lim_{x\to\infty}\frac{|h(x)-q(x)|}{x}<\infty$. This leads to novel applications regarding the uniform distribution of sequences of the from $h(\Omega(n))$, $h(\omega(n))$, and $h(\Omega(q_n))$. For example, we show that $(\Omega(n)^c)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$.

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.

Referee Report

1 major / 0 minor

Summary. The paper establishes a general theorem for estimating weighted averages E^W_{n≤N} f(θ(n)) for arbitrary f, under the assumption that the arithmetic function θ(n) satisfies a certain Gaussian distribution condition (with explicit mean/variance and error bounds sufficient for the weighted sums). It then applies this to derive an if-and-only-if criterion for the uniform distribution mod 1 of sequences (h(θ(n)))_{n∈ℕ} when h belongs to a Hardy field with polynomial growth: the sequence is u.d. mod 1 precisely when either (i) |h(x)−p(x)|/(x log x)→∞ for all rational polynomials p, or (ii) |h(x)−p(x)|/√x→∞ for all p and there exists q with |h(x)−q(x)|/x bounded. The result is stated for θ(n) equal to Ω(n), ω(n), or Ω(q_n) (the n-th square-free integer), and yields concrete corollaries such as the u.d. mod 1 of (Ω(n)^c) iff c is a non-integer >1/2.

Significance. If the Gaussian condition is verified for all listed θ, the work supplies a unified, quantitative framework that recovers and extends classical equidistribution results for additive arithmetic functions, while providing the first such criterion for Ω(q_n). The Hardy-field formulation and the sharp threshold involving x log x versus √x are technically novel and could serve as a template for further applications to other arithmetic functions satisfying Erdős–Kac-type statistics.

major comments (1)
  1. [Main theorem statement and applications to Ω(q_n)] The main weighted-average theorem (whose statement appears in the abstract and is invoked throughout the applications) requires θ(n) to satisfy a quantitative Gaussian condition with explicit mean ∼ log log n, variance ∼ log log n, and discrepancy bounds adequate for the weighted sums. While this is classical for Ω(n) and ω(n), the case θ(n)=Ω(q_n) is non-standard: q_n ∼ (π²/6)n and the prime factors of q_n are square-free by definition, so the additive function Ω(q_n) has a different generating process. The manuscript asserts the condition holds “similarly” for Ω(q_n) but does not derive the required mean, variance, or error terms for the weighted averages; this step is load-bearing for the equidistribution criterion applied to h(Ω(q_n)) and for the concrete example (Ω(q_n)^c).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the Gaussian condition in the case of Ω(q_n). We agree that this justification is essential for the applications and will strengthen the paper accordingly.

read point-by-point responses

  1. Referee: The main weighted-average theorem (whose statement appears in the abstract and is invoked throughout the applications) requires θ(n) to satisfy a quantitative Gaussian condition with explicit mean ∼ log log n, variance ∼ log log n, and discrepancy bounds adequate for the weighted sums. While this is classical for Ω(n) and ω(n), the case θ(n)=Ω(q_n) is non-standard: q_n ∼ (π²/6)n and the prime factors of q_n are square-free by definition, so the additive function Ω(q_n) has a different generating process. The manuscript asserts the condition holds “similarly” for Ω(q_n) but does not derive the required mean, variance, or error terms for the weighted averages; this step is load-bearing for the equidistribution criterion applied to h(Ω(q_n)) and for the concrete example (Ω(q_n)^c).

    Authors: We agree with the referee that the manuscript only indicates the condition holds similarly for Ω(q_n) without a full derivation of the mean, variance, and error bounds. In the revised version we will insert a new subsection (immediately following the statement of the main theorem) that derives these quantities explicitly. The argument proceeds by noting that the square-free integers q_n are the support of the density 6/π² and that Ω restricted to square-free arguments can be treated via the same probabilistic model as the classical Erdős–Kac theorem, adjusted by the Buchstab function and Selberg sieve estimates to control the contribution of small primes. This yields mean and variance both ∼ log log n (with an explicit additive constant arising from the density) together with discrepancy bounds of the same strength as those already used for Ω(n) and ω(n). The resulting estimates are then sufficient to apply the weighted-average theorem directly to h(Ω(q_n)), thereby justifying the equidistribution criterion and the concrete corollary for non-integer powers c > 1/2. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation flows from external Gaussian assumption to equidistribution criterion

full rationale

The paper's central weighted-average theorem is stated for any arithmetic function θ(n) satisfying an external Gaussian distribution condition (invoked as a hypothesis, with classical Erdős–Kac supplying it for Ω(n) and ω(n)). The equidistribution criterion for h(θ(n)) is then derived directly from this theorem plus standard Hardy-field properties; neither the criterion nor the theorem reduces by the paper's own equations to a fitted parameter, self-definition, or self-citation chain. The application to Ω(q_n) rests on an unverified extension of the Gaussian condition, but this is a correctness gap rather than circularity. No load-bearing step matches any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the arithmetic functions θ(n) satisfy a Gaussian distribution condition; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption θ(n) adheres to a certain Gaussian distribution condition
    Invoked to justify the general weighted-average theorem for θ = Ω, ω, Ω(q_n).

pith-pipeline@v0.9.0 · 5716 in / 1435 out tokens · 34606 ms · 2026-05-10T17:53:49.540263+00:00 · methodology

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Reference graph

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