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The variance of a general class of multiplicative functions in short intervals

This paper establishes asymptotic estimates for the variance of a general class of multiplicative functions in short intervals by connecting short and long averages through Fourier analysis and rational point counting, thereby improving existing results and disproving a conjecture by van Overbeeke regarding the variance of $\frac{\phi(n)}{n}$ .

Original authors: Pranendu Darbar, Mithun Kumar Das

Published 2026-06-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Pranendu Darbar, Mithun Kumar Das

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to understand the behavior of a mysterious crowd of numbers. These numbers aren't random; they follow strict rules based on their factors (like how many ways you can divide them). Mathematicians call these multiplicative functions.

For a long time, mathematicians have been good at counting these numbers over huge distances (like looking at the entire population of a city). But what happens if you only look at a tiny neighborhood (a "short interval")? Do the numbers behave the same way, or do they get chaotic?

This paper, written by Pranendu Darbar and Mithun Kumar Das, is like a new, high-tech microscope that allows us to see exactly how these numbers fluctuate in those tiny neighborhoods.

Here is the breakdown of their work using simple analogies:

1. The Big Question: The "Variance" of the Crowd

Imagine you are counting how many people in a crowd are wearing red hats.

The Long Average: If you look at the whole city, you might find that 10% of people wear red hats. This is the "global average."
The Short Interval: If you walk down just one street, you might find 15% or 5%.
The Variance: This paper measures how much the percentage on that one street jumps around compared to the city average. If the numbers are perfectly predictable, the variance is low. If they are chaotic, the variance is high.

The authors wanted to create a universal formula to predict this "jumpiness" (variance) for a whole family of these number-crowds.

2. The Tools: Fourier Analysis and "Binary Forms"

To solve this, the authors used two main tools:

Fourier Analysis: Think of this as a prism. Just as a prism breaks white light into a rainbow of colors, Fourier analysis breaks complex number patterns into simpler, wave-like frequencies. This helps them see the hidden rhythm in the chaos.
Counting Rational Points on Binary Forms: This sounds scary, but imagine a grid of dots. The authors had to count how many dots fall inside a specific, oddly shaped boundary (like a stretched-out oval). They used advanced geometry to count these dots efficiently, which helped them calculate the "noise" in the number sequences.

3. The Main Characters: The "General Class"

The authors didn't just study one specific type of number. They created a "General Class" (a big bucket) that holds many famous number functions:

The $k$ -free integers: Numbers that don't have any "perfect power" factors (like numbers that aren't divisible by 4, 8, 9, 16, etc.).
The Euler Totient Function ( $\phi(n)$ ): A function that counts how many numbers less than $n$ share no common factors with $n$ .
The Divisor Sums: Functions that add up the divisors of a number.

They proved that despite these functions looking different, they all follow the same underlying rules when you look at them in short intervals.

4. The Big Surprise: Breaking a Conjecture

One of the most exciting parts of the paper is a "plot twist."

The Old Theory: A mathematician named van Overbeeke had a theory (a conjecture) about how the Euler Totient function behaves. He thought that if you looked at short intervals, the "jumpiness" (variance) would settle down to a single, fixed constant number, no matter how big the interval got.
The New Discovery: Darbar and Das proved this wrong.
The Reality: They showed that the "jumpiness" doesn't settle on one fixed number. Instead, it oscillates (wiggles up and down) depending on the specific size of the interval.
- Analogy: Imagine a pendulum. Van Overbeeke thought the pendulum would eventually stop swinging and hang perfectly still. Darbar and Das showed that the pendulum keeps swinging, and the size of the swing depends on exactly how long you let it run.

5. Why Does This Matter?

The authors don't claim this will cure diseases or build better phones. Instead, they are refining the map of the number world.

They improved the range of "short intervals" where we can make accurate predictions.
They provided better formulas for how these numbers behave, which helps other mathematicians understand the deeper structure of numbers (like the distribution of prime numbers).
They corrected a specific misunderstanding about the Euler Totient function, showing that the behavior of numbers is more dynamic and less static than previously thought.

Summary

In short, this paper is a mathematical tour de force that uses advanced wave analysis and geometry to map out the "bumps and dips" of famous number sequences in small neighborhoods. They successfully built a universal guide for these patterns and, in the process, proved that a popular theory about one of these patterns was slightly off, revealing that the numbers are more lively and unpredictable than anyone expected.

Technical Summary: The Variance of a General Class of Multiplicative Functions in Short Intervals

Problem Statement
The paper addresses the asymptotic evaluation of the variance of multiplicative functions within short intervals. While the global behavior of multiplicative functions (specifically the partial sums $\sum_{n \le X} f(n)$ ) is well-studied, understanding their fluctuations in short intervals of length $H$ (where $H \ll X$ ) presents significant challenges. The authors focus on the continuous variance defined as:
$\text{Var}_f(X; H) := \frac{1}{X} \int_X^{2X} \left| \sum_{x < n \le x+H} f(n) - \frac{H}{X} \sum_{n \le X} f(n) \right|^2 dx$
The primary goal is to establish precise asymptotic formulas for this variance for a broad class of multiplicative functions, extending beyond specific cases like the Möbius function or square-free indicators to a general framework involving parameters for growth and order.

Methodology
The authors employ a combination of Fourier analysis, complex analysis, and the arithmetic geometry of binary forms. The methodology is structured around decomposing the multiplicative function $f(n)$ into a convolution structure $f = 1 *_k h$ , where $h$ belongs to a specific class of multiplicative functions.

Decomposition and Truncation: The sum over the short interval is split into "small divisors" (up to a parameter $z$ ) and "large divisors" (greater than $z$ ). The variance is analyzed by estimating the contribution of these two parts separately, denoted as $J_{h,z}(X; H)$ and $K_{h,z}(X; H)$ .
Fourier Analysis and Residue Calculus: For the small divisor part, the authors utilize Fourier analysis, specifically the Fourier expansion of Bernoulli polynomials and the function $S(x) = \frac{\sin \pi x}{\pi x}$ . They employ contour integration and the Cauchy residue theorem to evaluate integrals involving Dirichlet series and the Riemann zeta function. This allows them to extract the main terms and handle the poles of the associated generating functions.
Counting Rational Points on Binary Forms: A critical innovation in handling higher-order cases ( $k \ge 3$ ) is the use of counting rational points on specific binary forms. Instead of relying solely on Dirichlet approximation (used in previous works for $k=2$ ), the authors apply results from Heath-Brown, Stewart, and Xiao regarding the density of integer solutions to binary forms $F(x, y)$ . This technique is essential for bounding the off-diagonal terms in the variance calculation.
Large Value Theorems: For the large divisor part, the authors utilize Huxley's large value theorem, Bourgain's sub-convexity bound for the Riemann zeta function, and Montgomery's mean value theorem to control the error terms, particularly in the "unconditional" setting without assuming the Lindelöf hypothesis.

Key Contributions and Results

General Class Definition: The paper defines a general class of multiplicative functions $\mathcal{F}_{\alpha, \beta, k}$ characterized by parameters $\alpha$ (growth), $\beta$ (related to the magnitude of values on primes), and $k$ (order). This class encompasses functions such as $\mu_k(n)$ (indicator of $k$ -free integers), $\phi(n)/n$ , and generalized divisor functions $\sigma_\alpha(n)$ .
Asymptotic Formulas for Variance:
- Case $0 \le \alpha < 1/2$ : The authors derive an asymptotic formula where the variance scales as $H^{(1-2\alpha)/k} (\log H)^{\beta^2-1}$ . The main term involves a constant $c_{h,k}$ and a polynomial in $\log H$ .
- Case $1/2 < \alpha < 2$ : The variance converges to a constant $c_{h,k}(H)$ that depends on the interval length $H$ , with an error term of $O(H^{-\epsilon})$ .
- Critical Point $\alpha = 1/2$ : The paper notes that a direct asymptotic formula is not derived for this critical case, though an upper bound of $O((\log H)^{\beta^2})$ is established.
Improved Ranges for Short Intervals: The paper significantly extends the known ranges of $H$ $H$ for which these asymptotic formulas hold.
- For $k$ -free integers ( $k \ge 3$ ), the valid range for $H$ is extended from approximately $X^{0.3}$ to $X^{e(k,0)}$ , where $e(k,0)$ increases with $k$ (e.g., $e(3,0) \approx 0.605$ ).
- Under the Lindelöf hypothesis, these ranges are further extended to $X^{g(k)}$ .
Disproof of a Conjecture: The authors provide a detailed analysis of the normalized Euler totient function $\phi(n)/n$ . They demonstrate that the variance limit is not an absolute constant as conjectured by van Overbeeke, but rather a function $c_\zeta(H)$ that fluctuates with $H$ . Specifically, they show that for prime $H$ , the constant is bounded below by $\approx 0.1036$ , contradicting the conjectured value of $\approx 0.03$ .
Generalized Divisor Functions: The results improve upon previous bounds for the variance of $\sigma_\alpha(n)$ for $\alpha \in (-1, -1/2)$ , providing asymptotic formulas over a wider range of $H$ and surpassing the $1/2$ -barrier for certain values of $\alpha$ .

Significance
The paper claims significance in unifying the study of variance for diverse multiplicative functions under a single theoretical framework. By connecting "short averages" to "long averages" through Fourier analysis and the geometry of numbers (binary forms), the authors provide a robust method for analyzing fluctuations in short intervals.

The work is particularly notable for:

Refuting the van Overbeeke Conjecture: It corrects a specific prediction regarding the variance of $\phi(n)/n$ , showing that the limit depends on the arithmetic properties of the interval length $H$ rather than being a universal constant.
Extending the "Half-Barrier": The methodology allows for the analysis of variance in intervals longer than $X^{1/2}$ for specific classes of functions, a regime where previous methods often failed or yielded weaker bounds.
Discrete vs. Continuous Variance: The paper distinguishes between discrete and continuous variances, noting that while the continuous variance can overcome the $1/2$ -barrier in Fourier space, the discrete variance remains bounded by it, highlighting a fundamental difference in the detection of small interval fluctuations.

The authors emphasize that their results yield new asymptotic formulas and improvements in the realm of short interval analysis for functions including $k$ -free indicators, the normalized Euler totient, and generalized divisor sums, without relying on unproven conjectures like the Riemann Hypothesis for the primary unconditional results, though RH and Lindelöf are discussed for potential range extensions.