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On the series expansion of the prime zeta function about $s=1$ and its coefficients

This paper derives a series expansion of the prime zeta function around its logarithmic singularity at $s=1$ , establishes a general formula for its coefficients analogous to the Stieltjes constants of the Riemann zeta function, and provides high-precision numerical verification of these results.

Original authors: Artur Kawalec

Published 2026-03-24

📖 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 trying to understand the rhythm of a chaotic drumbeat. The drummers are the prime numbers (2, 3, 5, 7, 11...), and they are scattered across the number line in a way that seems random, yet follows a hidden, deep pattern.

Mathematicians have a special tool called the Prime Zeta Function (let's call it $P(s)$ ) to listen to this rhythm. It's like a microphone that picks up the "sound" of all prime numbers at once. However, there's a problem: right at the center of the number line (at the point $s=1$ ), the microphone breaks. It screams with a "logarithmic singularity"—a mathematical way of saying the sound gets infinitely loud and messy, like a speaker blowing out.

This paper, written by Artur Kawalec, is about fixing that broken microphone and figuring out exactly how it breaks, so we can understand the music underneath.

Here is the breakdown of the paper using simple analogies:

1. The Problem: The "Blown Speaker" at $s=1$

Imagine you are trying to describe the sound of a storm. If you stand right in the eye of the hurricane ( $s=1$ ), the wind is so strong you can't hear anything else. The Prime Zeta Function behaves the same way at $s=1$ . It has a "singularity."

The author's goal is to take that messy, screaming noise, peel it away, and look at the smooth, regular pattern that remains underneath. He does this by writing a "recipe" (a series expansion) that describes the function as:

Total Sound = The Screaming Noise + The Smooth Background Music

The "Screaming Noise" is a known mathematical term ( $\log(\frac{1}{s-1})$ ). The "Smooth Background Music" is what the author is interested in. He wants to find the specific notes (coefficients) that make up that background music.

2. The Recipe: Finding the "Coefficients"

In math, when you break a complex curve down into a simple list of numbers (a series), those numbers are called coefficients. Think of them as the ingredients in a cake recipe.

The Old Way (Slow Cooking): The author first shows how to calculate these ingredients by adding up prime numbers one by one. But this is like trying to bake a cake by gathering every single grain of wheat from a field by hand. It works, but it takes forever and is very slow.
The New Way (The Fast Oven): The author discovers a shortcut. He realizes that the Prime Zeta Function is secretly connected to the famous Riemann Zeta Function (a more general version of the prime function) through a mathematical "magic trick" called Möbius inversion.
- Analogy: Instead of gathering wheat by hand, he realizes he can just buy a pre-mixed flour bag (the Riemann Zeta function) that already contains the prime information, but organized in a way that is much faster to process.

3. The Connection to "Mertens' Theorems"

You might have heard of Mertens' Theorems. These are famous rules about how prime numbers behave.

The author shows that his new "ingredients" (the coefficients) are actually just generalized versions of Mertens' old rules.
If Mertens' Theorems are like learning the basic scales on a piano, this paper teaches you how to play complex jazz improvisations using those same scales. It extends the old rules to higher, more complex levels of detail.

4. The "Ghost" in the Machine (The Remainder)

To prove his recipe works, the author looks at the "error" or the "remainder" when we try to predict where prime numbers are.

He compares the actual count of primes ( $\pi(x)$ ) to a smooth curve called the Logarithmic Integral ($li(x)$).
The difference between the real primes and the smooth curve is like the "static" or "noise" in a radio signal.
The author proves that if you integrate (sum up) this "static" noise in a very specific way, you get the exact numbers (coefficients) needed for his recipe.

5. The Results: A New Table of Constants

The paper doesn't just talk about theory; it actually does the math. The author calculated the first 10 "ingredients" (coefficients) to incredible precision.

He created a Reference Table (Table 1 in the paper) listing these numbers.
These numbers are like a new set of mathematical constants (similar to $\pi$ or $e$ ) that describe the deep structure of prime numbers.
For example, the first number in his list is roughly -0.3157. This number tells us exactly how the Prime Zeta Function behaves right next to the "broken speaker" at $s=1$ .

Summary: Why Does This Matter?

Think of the Prime Zeta Function as a map of a foggy mountain range.

Before this paper: We knew there was a foggy peak at $s=1$ where we couldn't see anything.
This paper: The author has drawn a detailed map of the fog itself. He figured out exactly how the fog swirls and how the terrain looks just underneath it.

By understanding these "swirls" (the coefficients), mathematicians can:

Predict prime number behavior more accurately.
Connect the Prime Zeta function to the Riemann Hypothesis (the biggest unsolved problem in math) more deeply.
Generalize old rules (Mertens' Theorems) to solve harder, more complex problems.

In short, Kawalec took a messy, broken mathematical function, cleaned it up, and gave us a precise instruction manual for its behavior, allowing us to hear the "music" of the primes much more clearly.

1. Problem Statement

The Prime Zeta Function, defined as $P(s) = \sum_{p} p^{-s}$ (where $p$ denotes prime numbers), converges absolutely for $\Re(s) > 1$ . However, via Möbius inversion, it can be analytically continued to $\Re(s) > 0$ , revealing that $P(s)$ possesses logarithmic singularities (branch points) at $s = 1/n$ and at the non-trivial zeros of the Riemann zeta function.

The central problem addressed in this paper is the derivation of a series expansion of $P(s)$ about the singularity at $s=1$ . Unlike the Riemann zeta function $\zeta(s)$ , which has a simple pole at $s=1$ , the prime zeta function has a logarithmic singularity. The paper aims to:

Define a specific branch of $P(s)$ via a series expansion around $s=1$ .
Derive general formulas for the expansion coefficients ( $\alpha_n$ ).
Connect these coefficients to known constants (like the Meissel-Mertens constant and Stieltjes constants) and provide high-precision numerical values.

2. Methodology

The author employs a combination of analytic number theory, complex analysis, and integral transforms to achieve the results.

Series Definition: The paper defines the expansion of $P(s)$ near $s=1$ as:
$P(s) = \log\left(\frac{1}{s-1}\right) + \sum_{n=0}^{\infty} \frac{\alpha_n}{n!} (s-1)^n$
This separates the logarithmic singularity from the regular analytic part.
Möbius Inversion: The function is expressed using the relationship with the Riemann zeta function:
$P(s) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \log \zeta(ns)$
This representation is crucial for deriving alternative formulas for the coefficients that converge much faster than direct summation over primes.
Stieltjes Integration: The core of the proof involves using the prime counting function $\pi(x)$ and the logarithmic integral $\text{li}(x)$ . By expressing sums over primes as Stieltjes integrals ( $\int d\pi(t)$ ) and utilizing the remainder term $f(x) = \pi(x) - \text{li}(x)$ , the author derives integral representations for the coefficients.
- The remainder $f(x)$ is estimated as $O(\sqrt{x} \log x)$ under the Riemann Hypothesis (RH).
Differentiation and Limits: The coefficients $\alpha_n$ are derived by taking limits of specific sums involving powers of logarithms of primes, analogous to the definition of Stieltjes constants for $\zeta(s)$ .

3. Key Contributions and Theorems

The paper presents three main theorems:

Theorem 1: Limit Formulas for Coefficients
The author derives explicit limit formulas for the coefficients $\alpha_n$ based on sums over primes:

For $n=0$ :
$\alpha_0 = \lim_{x \to \infty} \left( \sum_{p \le x} \frac{1}{p} - \log \log x \right) - \gamma$
This relates $\alpha_0$ to the Meissel-Mertens constant ( $M$ ) and Euler's constant ( $\gamma$ ), specifically $\alpha_0 = M - \gamma$ .
For $n \ge 1$ :
$\alpha_n = \lim_{x \to \infty} (-1)^n \left( \sum_{p \le x} \frac{\log^n p}{p} - \frac{\log^n x}{n} \right)$
These formulas generalize Mertens' Theorems to higher orders.

Theorem 2: Representation via Zeta Derivatives
A more computationally efficient formula is provided, expressing $\alpha_n$ in terms of the derivatives of $\log \zeta(s)$ at $s=1$ :
$\alpha_n = g_n + \frac{d^n}{ds^n} \left[ \sum_{k=2}^{\infty} \frac{\mu(k)}{k} \log \zeta(ks) \right]_{s \to 1}$
Here, $g_n$ are the expansion coefficients of $\log \zeta(s)$ about $s=1$ , which are expressible in terms of Stieltjes constants ( $\gamma_k$ ). This method allows for high-precision computation because the series involving the Möbius function converges rapidly.

Theorem 3: Integral Representation
An alternative integral formula for the coefficients is derived, connecting them directly to the error term in the Prime Number Theorem ( $f(t) = \pi(t) - \text{li}(t)$ ):
$\alpha_n = (-1)^n \int_1^{\infty} \frac{\log^n t - n \log^{n-1} t}{t^2} [\pi(t) - \text{li}(t)] \, dt$
This provides a theoretical link between the expansion coefficients and the distribution of prime numbers.

4. Results

Generalization of Mertens' Theorems: The paper successfully generalizes the classical results of Mertens (which deal with $n=0$ and $n=1$ ) to arbitrary orders $n$ .
High-Precision Numerical Values: Using the fast-converging formula from Theorem 2, the author computed the first 10 coefficients ( $\alpha_0$ $α_{0}$ through $\alpha_{10}$ $α_{10}$ ) to 30 decimal places.
- Example: $\alpha_0 \approx -0.315718...$
- Example: $\alpha_1 \approx 1.332582...$
- The coefficients grow rapidly in magnitude and alternate in sign.
Analytic Continuation: The paper establishes that the regular part of the series is analytic on a disk of radius $R=1/2$ centered at $s=1$ , with a branch cut defined on $(1/2, 1]$ .

5. Significance

Computational Utility: The derived formulas (specifically Theorem 2) offer a practical method for computing these constants to high precision, which was previously difficult due to the slow convergence of direct prime summation.
Theoretical Insight: The work bridges the gap between the distribution of primes (via $\pi(x)$ and $\text{li}(x)$ ) and the analytic properties of the Prime Zeta function. It demonstrates that the coefficients of the singularity expansion are deeply rooted in the Stieltjes constants of the Riemann zeta function and the error term of the Prime Number Theorem.
Foundation for Further Research: By providing a rigorous framework for the expansion of $P(s)$ , this paper lays the groundwork for future studies on the behavior of prime zeta functions near singularities and their potential applications in analytic number theory.

In summary, Kawalec provides a comprehensive analytical and numerical treatment of the Prime Zeta function's behavior near $s=1$ , offering new general formulas for its coefficients and verifying them with high-precision data.

1. The Problem: The "Blown Speaker" at s=1s=1s=1