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On the series expansion of k-free Dirichlet series and its analytical continuation

This paper develops a Laurent series expansion with a simple pole for the $k$ -free zeta Dirichlet series, establishes a Stieltjes-like formula for its regular part coefficients, and derives a new analytical continuation that expresses $\zeta(1/k)$ in terms of the $k$ -free indicator function.

Original authors: Artur Kawalec

Published 2026-03-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Artur Kawalec

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 hidden rhythm of numbers. In the world of mathematics, there is a famous "song" called the Riemann Zeta function ( $\zeta(s)$ ). This song is beautiful, but it has a very loud, chaotic note (a "pole") at the number 1 that makes it impossible to sing clearly right there.

For a long time, mathematicians have been trying to figure out how to "tune" this song, especially for specific types of numbers. This paper by Artur Kawalec is like a new sheet of music that helps us understand a specific group of numbers called "k-free" numbers.

Here is the breakdown of what the paper does, using simple analogies:

1. The "k-free" Numbers: The Uncluttered Room

First, let's understand what a "k-free" number is.

Imagine you have a room full of furniture (numbers).
Some pieces of furniture are huge, bulky cubes (perfect squares like $4, 9, 16$).
Some are even bigger cubes (perfect cubes like $8, 27, 64$).
A square-free number is a room where you have removed all the bulky cubes. You only have items that don't fit perfectly into a square box.
A k-free number is a room where you've removed all items that fit perfectly into a "k-dimensional" box.

The author is studying the "music" (the Dirichlet series) made specifically by these uncluttered, k-free numbers.

2. The Big Problem: The Loud Note at 1

The main character of this story is a mathematical formula that combines the famous Zeta song with a filter (dividing by $\zeta(ks)$ ).

The Issue: When you try to listen to this formula at the number $s=1$ , it screams. It blows up. It's like trying to listen to a radio station right next to a massive speaker; the signal is too loud to hear the details.
The Solution: The author figures out how to "mute" that loud scream (the pole) and listen to the quiet, beautiful melody underneath. He does this by creating a Laurent Series.
- Analogy: Imagine a stormy sea (the formula). The author builds a boat (the series expansion) that floats right over the biggest wave (the pole) so you can see the calm water underneath.

3. The "Stieltjes" Recipe: Measuring the Waves

To understand the quiet melody under the storm, the author uses a technique called Stieltjes integration.

Analogy: Imagine you are trying to measure the weight of a pile of sand, but the sand keeps shifting. Instead of weighing it all at once, you take a scoop, measure it, and then look at how the pile changes as you add more sand.
The author uses this method to find the "coefficients" of the melody. These coefficients are like the specific notes in a chord. He proves a formula (Theorem 1) that tells us exactly how to calculate these notes by looking at how the k-free numbers behave as they get larger and larger.

4. The "Euler" Constant: The New Standard

In math, there is a famous number called the Euler-Mascheroni constant ( $\gamma \approx 0.577$ ). It's a "universal constant" that appears whenever you add up fractions like $1 + 1/2 + 1/3 + \dots$ .

The author discovers that for k-free numbers, there is a new version of this constant, which he calls $\gamma_{M,k}$ .
Analogy: Think of the original Euler constant as the "standard gravity" on Earth. The author found that if you live on a different planet (a k-free world), the gravity is slightly different.
The Cool Discovery: As you make the "k" bigger and bigger (removing more and more bulky furniture), this new gravity constant slowly drifts back toward the original Earth gravity. It's like the k-free world slowly becoming more and more like our normal world.

5. The "Time Machine" for Zeros

The paper also looks at a very strange place: the number $1/k$ .

Usually, the Zeta function has "zeros" (places where the song stops) at specific negative numbers. But when you look at the k-free version, a new zero appears at $1/k$ .
The author creates a formula to calculate the value of the Zeta function at these tricky spots ( $\zeta(1/k)$ ).
Analogy: Imagine you have a broken clock that only works at 12:00. The author built a time machine that lets you peek at what the clock would say if it worked at 1:00, 2:00, etc., by using a special mathematical trick involving the k-free numbers.

6. The Visual Proof: The Wiggly Line

Finally, the author didn't just do the math on paper; he ran computer simulations.

He plotted these formulas on a graph.
Analogy: Imagine drawing a line that is supposed to go straight to a target (the true answer). But because the numbers are so chaotic, the line wiggles and shakes wildly like a drunk person trying to walk straight.
The Result: Even though the line wiggles, it slowly settles down and hits the target. The author shows that as you look at more and more numbers (going further down the line), the shaking gets smaller, and the line finally lands exactly on the correct value of $\zeta(1/k)$ .

Summary

In short, this paper is a guidebook for navigating a chaotic mathematical landscape. It teaches us how to:

Filter out the "noise" (the pole) to hear the music.
Find new constants that describe the rhythm of special numbers.
Calculate values of famous functions at places where they usually break.

It's a bit like taking a broken, noisy radio, building a new antenna, and finally hearing the clear, beautiful song of the numbers underneath.

1. Problem Statement

The paper addresses the analytical properties of the $k$ -free Dirichlet series, defined as:
$\frac{\zeta(s)}{\zeta(ks)} = \sum_{n=1}^{\infty} \frac{\mu^{(k)}(n)}{n^s}$
where $\mu^{(k)}(n)$ is the indicator function for $k$ -free integers (integers not divisible by any $k$ -th power $>1$ ).

While the case for square-free integers ( $k=2$ ) has been previously studied (specifically regarding Laurent expansions and constants analogous to Euler's constant), this work aims to generalize these results for any integer $k \geq 2$ . The specific objectives are:

To derive the Laurent series expansion of the series around the pole $s=1$ .
To establish a Stieltjes-like limit formula for the expansion coefficients of the regular part.
To investigate an alternative analytical continuation of the series.
To derive explicit formulas for $\zeta(1/k)$ in terms of $k$ -free indicator functions.

2. Methodology

The author employs a combination of analytic number theory techniques, specifically:

Dirichlet Series and Indicator Functions: Utilizing the relation $\mu^{(k)}(n) = \sum_{d^k|n} \mu(d)$ to define the series.
Laurent Series Expansion: Expanding the function $\frac{\zeta(s)}{\zeta(ks)}$ around the simple pole at $s=1$ .
Stieltjes Integration: Transforming discrete sums into integrals involving the $k$ -free counting function $Q_k(x) = \sum_{n \leq x} \mu^{(k)}(n)$ .
Asymptotic Analysis: Using the asymptotic behavior $Q_k(x) \sim \frac{x}{\zeta(k)}$ and the error term $f_k(x) = Q_k(x) - \frac{x}{\zeta(k)}$ . The paper notes that under the Riemann Hypothesis (RH), the error term is bounded by $O(x^{1/2k})$ .
Differentiation and Limit Processes: Differentiating the series representation to extract coefficients and taking limits as $x \to \infty$ to isolate constants.

3. Key Contributions and Results

A. Laurent Series Expansion and Coefficients

The paper establishes the Laurent expansion of the $k$ -free Dirichlet series around $s=1$ :
$\frac{\zeta(s)}{\zeta(ks)} = \frac{1}{\zeta(k)(s-1)} + \sum_{n=0}^{\infty} \frac{\gamma_{M,k}^n}{n!} (s-1)^n$
The radius of convergence is determined to be $R = 1 + \frac{2}{k}$ , limited by the trivial zeros of $\zeta(ks)$ .

Theorem 1 (Generalized Wolf Formula):
The author derives a general log-limit formula for the coefficients $\gamma_{M,k}^n$ of the regular part:
$\gamma_{M,k}^n = \lim_{x \to \infty} (-1)^n \left( \sum_{j=1}^{x} \frac{\mu^{(k)}(j) \log^n(j)}{j} - \frac{\log^{n+1}(x)}{\zeta(k)(n+1)} \right)$
This generalizes a known result by Wolf for the $k=2$ (square-free) case.

B. Generalized Euler Constant ( $\gamma_{M,k}$ )

For the $n=0$ case, the paper defines a generalized constant $\gamma_{M,k}$ (analogous to the Euler-Mascheroni constant $\gamma$ ):
$\gamma_{M,k} = \lim_{x \to \infty} \left( \sum_{n \leq x} \frac{\mu^{(k)}(n)}{n} - \frac{1}{\zeta(k)} \log x \right)$
The paper provides a closed-form expression for this constant:
$\gamma_{M,k} = \frac{\gamma}{\zeta(k)} - \frac{k \zeta'(k)}{\zeta(k)^2}$
Numerical Findings:

High-precision values for $\gamma_{M,k}$ are computed for $k=2$ to $11$.
The sequence converges to the standard Euler constant as $k \to \infty$ : $\lim_{k \to \infty} \gamma_{M,k} = \gamma$ .
Tables 2–5 provide the first 10 higher-order coefficients ( $\gamma_{M,k}^n$ ) for $k=2,3,4,5$ with 30 decimal places of precision.

C. Alternative Analytical Continuation

The paper derives an analytical continuation valid for $\Re(s) > \frac{1}{2k}$ (assuming RH), expressed as:
$\frac{\zeta(s)}{\zeta(ks)} = \lim_{x \to \infty} \left( \sum_{n \leq x} \frac{\mu^{(k)}(n)}{n^s} - \frac{x^{1-s}}{\zeta(k)(1-s)} \right)$
This formula serves as an intermediate step between the standard Dirichlet series and the Euler-Maclaurin continuation of the Riemann zeta function.

D. Formulas for $\zeta(1/k)$

By analyzing the behavior of the series near the pole of the denominator at $s = 1/k$ , the author derives limit formulas to compute $\zeta(1/k)$ :
$\zeta\left(\frac{1}{k}\right) = \lim_{x \to \infty} \left( -\frac{1}{k} \sum_{n \leq x} \frac{\mu^{(k)}(n)(\log n + \frac{k}{k-1})}{n^{1/k}} + \frac{1}{\zeta(k)(k-1)} x^{1-1/k} \log x \right)$
Numerical verification is provided for $k=2, 3, 4, 5$ , showing that the series oscillates around the true value of $\zeta(1/k)$ with decaying amplitude.

4. Significance

Generalization: The work successfully extends specific results known for square-free integers ( $k=2$ ) to the general $k$ -free case, unifying the theory of these Dirichlet series.
New Constants: It introduces and computes a family of constants $\gamma_{M,k}$ , linking them to the derivative of the Riemann zeta function and the standard Euler constant.
Computational Utility: The derived limit formulas for $\zeta(1/k)$ offer a novel method for computing values of the zeta function at negative integers (via $1/k$ ) using $k$ -free indicators, which is relevant for numerical analysis and verification of zeta values.
Theoretical Insight: The derivation of the analytical continuation radius and the connection to the Riemann Hypothesis (via the error term bounds) deepens the understanding of the distribution of $k$ -free numbers.

In summary, the paper provides a rigorous analytical framework for $k$ -free Dirichlet series, offering explicit formulas for their expansion coefficients, generalized constants, and values at specific points, supported by high-precision numerical data.