Weighted averages of arithmetic functions and applications to equidistribution

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 $\vartheta(n)=Ω(n)$, $\vartheta(n)=ω(n)$, or $\vartheta(n)=Ω(q_n)$, where $Ω(n)$ and $ω(n)$ 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 $(h(\vartheta(n)))<em>{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if one of the following (mutually exclusive) conditions is satisfied: (i) $$\lim</em>{x\to\infty} \frac{|h(x)-p(x)|}{x \log x}=\infty$$ 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 $\lim_{x\to\infty}\frac{|h(x)-q(x)|}{x}&lt;\infty$. This leads to novel applications regarding the uniform distribution of sequences of the from $h(Ω(n))$, $h(ω(n))$, and $h(Ω(q_n))$. For example, we show that $(Ω(n)<sup>c)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$.

• The paper introduces a unified framework for estimating weighted averages of arithmetic functions via discrete and binomial means.
• It establishes sharp asymptotic equivalences between classical Cesàro averages and parity-neutral binomial means under Gaussian local distribution conditions.
• The paper characterizes uniform distribution mod 1 for Hardy field functions, linking equidistribution properties across different arithmetic sequences.

Weighted Averages of Arithmetic Functions and Applications to Equidistribution

Overview and Main Results

The paper "Weighted averages of arithmetic functions and applications to equidistribution" (2604.07186) develops a comprehensive theory for estimating weighted averages of compositions $f(\vartheta(n))$, where $f$ is an arbitrary complex-valued function, and $\vartheta(n)$ is an arithmetic function (such as $\Omega(n)$ or $\omega(n)$) whose statistics locally resemble a Gaussian distribution. The work introduces a unified framework for analyzing averages with respect to discrete weights, binomial means, and their parity-neutral variants. The main theorems establish precise asymptotic relationships between weighted averages of arithmetic functions and binomial means, and provide sharp criteria for uniform distribution mod $1$ of sequences parametrized by Hardy field functions.

Weighted Averages and Binomial Means

For weights $W: \mathbb{N}\to [0,\infty)$, the authors define discrete averages of the form

$$^{W}_{n \le N} f(\vartheta(n)) = \frac{1}{W(N)} \sum_{n=1}^N \Delta W(n) f(\vartheta(n)),$$

with $\Delta W$ denoting the discrete derivative, and $f$ arbitrary. Binomial and parity-neutral binomial means are also considered: $f$0

The principal technical result asserts, under mild regularity and growth conditions for $f$1 and for arithmetic functions $f$2 admitting a Gaussian local distribution, that

$f$3

where $f$4 is a sublinear function associated with the mean and variance of $f$5. This forms the basis for a hierarchy of averaging results at different scales (Cesàro, logarithmic, double-logarithmic), and generalizes several classical number-theoretic results by relating averages along $f$6 or $f$7 to binomial means of shifted arguments.

Gaussian Condition and Applications

The scope of the framework covers arithmetic functions $f$8 whose distribution among $f$9 closely approximates a normal law, such as:

• $\vartheta(n)$0: the total number of prime factors of $\vartheta(n)$1, counting multiplicities
• $\vartheta(n)$2: the number of distinct prime factors of $\vartheta(n)$3
• $\vartheta(n)$4: the $\vartheta(n)$5 function evaluated at the $\vartheta(n)$6-th squarefree integer

For these, the local distribution is essentially governed by $\vartheta(n)$7 and $\vartheta(n)$8 variance. The paper's Gaussian condition formalizes this approximation and forms the backbone of the weighted averaging theory.

Criteria for Uniform Distribution mod $\vartheta(n)$9 for Hardy Field Functions

A significant application is to uniform distribution mod $\Omega(n)$0 of sequences $\Omega(n)$1 where $\Omega(n)$2 is a Hardy field function (which excludes highly oscillatory behavior and includes all polynomials and logarithmico-exponential functions). The authors provide a dichotomy (Theorem \ref{thm_ud_of_Hardy_functions_along_Omega_Cesaro}):

• $\Omega(n)$3 is uniformly distributed mod $\Omega(n)$4 if and only if either
  • $\Omega(n)$5 for all $\Omega(n)$6,
  • or $\Omega(n)$7 for each $\Omega(n)$8, and there exists $\Omega(n)$9 such that $\omega(n)$0.

This precise characterization leads to strong, contradicting statements, such as: $\omega(n)$1 is uniformly distributed mod $\omega(n)$2 if and only if $\omega(n)$3 is a non-integer strictly greater than $\omega(n)$4.

Double-Logarithmic Averaging and Equivalence of Distribution

The authors demonstrate that, with double-logarithmic averages, uniform distribution mod $\omega(n)$5 of $\omega(n)$6, $\omega(n)$7 (where $\omega(n)$8 is the $\omega(n)$9-th prime), and $1$0 are equivalent. This result ties weighted averages along arithmetic functions directly to classical uniform distribution properties, revealing new connections between ergodic theory, arithmetic combinatorics, and asymptotic prime factor statistics.

Strong Numerical and Theoretical Results

• Binomial means achieve sharp error bounds and precise equivalence to Cesàro averages along $1$1, reconciling expected values with known results such as the Prime Number Theorem by showing $1$2.
• Distribution mod $1$3: For Hardy field functions $1$4 of polynomial growth, the paper establishes necessary and sufficient conditions for the sequence $1$5 to be uniformly distributed mod $1$6—demonstrating subtle distinctions that are not visible in classical, unweighted settings.
• Equivalence of double-log averages: Uniform distribution mod $1$7 along $1$8, along the sequence of primes, and along $1$9 become equivalent with double-logarithmic weighting.

Implications and Speculation

The theoretical results bridge analytic number theory, uniform distribution, and ergodic theory. Practically, the weighted averaging framework offers a robust toolset for researchers who require precise asymptotics for arithmetic functions with non-classical averaging mechanisms and for the study of distribution phenomena in number-theoretic and dynamical contexts. The binomial mean approach, by capturing local normality in arithmetic statistics, may serve as a template for further generalizations to non-standard sequences (e.g., squarefree, k-almost primes, or values along polynomially parametrized sets).

Future developments may extend the scope to multi-dimensional arithmetic functions, finer analytic weights, or potential applications within probabilistic number theory and ergodic dynamical systems. The interplay between Hardy field phenomena and arithmetic statistics sets the stage for further investigation of polynomial and transcendental rates in uniform distribution, as well as for novel Tauberian theorems relating distributional properties under non-classical weighting.

Conclusion

This paper provides a rigorous, highly technical theory of weighted averages of arithmetic functions, establishing close relationships to binomial means and uniform distribution phenomena. Its results yield sharp dichotomies and equivalences, with direct implications for classical problems in number theory, ergodic theory, and uniform distribution. The work sets a high standard for analytical precision and opens avenues for further exploration in the asymptotic analysis of arithmetic functions.