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Analytical continuation of prime zeta function for $\Re(s)>\frac{1}{2}$ assuming (RH)

Assuming the Riemann Hypothesis, this paper derives a simple expression for the analytic continuation of the prime zeta function to the domain $\Re(s)>\frac{1}{2}$ , verifies the formula numerically, and provides illustrative plots.

Original authors: Artur Kawalec

Published 2026-03-27

📖 4 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 map a mysterious, foggy landscape called the Prime Zeta Function. This landscape is built entirely from prime numbers (2, 3, 5, 7, etc.).

For a long time, mathematicians knew how to walk safely on the "high ground" where the numbers are large (specifically, where the real part of the variable $s$ is greater than 1). But as they tried to walk toward the "danger zone" closer to zero, the path became foggy and eventually hit a cliff.

Here is the story of what this paper does, explained simply:

1. The Problem: The Foggy Cliff

Think of the Prime Zeta Function as a bridge made of prime numbers.

The Old Bridge: We had a way to cross the bridge for the first half (where $s > 1$ ).
The Extension: Later, mathematicians found a clever trick (using something called the Möbius inversion) to extend the bridge a bit further, down to $s > 0$ .
The Wall: However, right at the edge of zero, there is a "natural boundary." It's like a wall of dense, chaotic fog. The bridge seems to crumble into an infinite number of tiny cracks, making it impossible to cross further using old methods.

2. The New Discovery: A Magic Compass

The author, Artur Kawalec, claims to have found a new compass that allows us to walk even further, down to the halfway point ( $s > 1/2$ ), if we assume a famous hypothesis called the Riemann Hypothesis (RH) is true.

Think of the Riemann Hypothesis as a "Golden Rule" of the universe of numbers. If we assume this rule holds, the chaotic fog near the cliff organizes itself into a predictable pattern.

3. The Solution: The "Exponential Integral" Tool

The paper introduces a new formula. To understand it, imagine you are trying to count the total weight of all the prime numbers up to a certain point.

The Direct Count: You add them up one by one. This works well when you are far away, but as you get closer to the danger zone, the sum gets messy and unstable.
The Correction: The author says, "Don't just add them up. Add them up, and then apply a magic correction factor."

This correction factor is a special mathematical tool called the Exponential Integral (denoted as $E_1$ ).

The Analogy: Imagine you are driving a car toward a steep hill. If you just keep the gas pedal down, you might crash. But if you have a special "brake and steer" system (the $E_1$ function) that automatically adjusts your speed and direction based on how close you are to the edge, you can drive right up to the cliff's edge without falling off.

The paper proves that if you take the sum of primes and add this specific "magic correction," the math stays smooth and stable all the way down to the $1/2$ mark.

4. The Catch: The "Branch Cut"

There is one small rule you have to follow. The paper mentions a "branch cut."

The Metaphor: Imagine the landscape is a spiral staircase. If you walk around the center too many times, you might end up on a different floor than where you started, even though you didn't go up or down.
To keep things simple, the author draws a "No-Go Zone" (a line from 0.5 to 1) on the map. As long as you stay on one side of this line, the math works perfectly. If you try to cross it, the numbers get confused.

5. The Proof: The Computer Test

The author didn't just write down the theory; they wrote a computer program (using a language called Pari/GP) to test it.

They compared the "Old Bridge" (the known formula) with their "New Compass" (the new formula).
The Result: The two maps matched perfectly! Whether they looked at the real numbers or the imaginary parts, the new formula reproduced the old one exactly, proving the math works.

Summary

In short, this paper says:

"We found a new way to extend the map of prime numbers. By assuming the Riemann Hypothesis is true and using a special 'correction tool' (the Exponential Integral), we can safely explore the territory down to the halfway point ( $s > 1/2$ ), which was previously thought to be too foggy to navigate."

It's like finding a new pair of glasses that lets you see clearly through a fog that everyone else thought was impenetrable.

Based on the provided paper "Analytical continuation of prime zeta function for $\Re(s) > \frac{1}{2}$ assuming (RH)" by Artur Kawalec, here is a detailed technical summary:

1. Problem Statement

The Prime Zeta Function, defined as $P(s) = \sum_{p} p^{-s}$ , is a fundamental object in analytic number theory.

Current Limitations: The series converges absolutely only for $\Re(s) > 1$ . While the Möbius inversion formula, $P(s) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \log \zeta(ns)$ , extends the domain to $\Re(s) > 0$ , the function encounters a natural boundary at $\Re(s) = 0$ . Beyond this line, $P(s)$ cannot be analytically continued due to a dense accumulation of singularities.
The Gap: There is a need for a representation that allows the analytic continuation of $P(s)$ into the critical strip, specifically for the domain $\frac{1}{2} < \Re(s) \le 1$ , under the assumption of the Riemann Hypothesis (RH).

2. Methodology

The author employs Stieltjes integration combined with the asymptotic properties of the prime counting function $\pi(x)$ .

Prime Counting Function: The proof utilizes the definition of $\pi(x)$ and its asymptotic expansion:
$\pi(x) = \text{li}(x) + f(x)$
where $\text{li}(x)$ is the logarithmic integral and $f(x)$ is the remainder term.
Assumption of RH: Under the Riemann Hypothesis, the error term is bounded by $f(x) = O(\sqrt{x} \log x)$ .
Integration by Parts: The author expresses $P(s)$ $P (s)$ as a Stieltjes integral involving $d\pi(t)$ $d π (t)$ . By substituting $\pi(t) = \text{li}(t) + f(t)$ $π (t) = li (t) + f (t)$ , the integral is split into a main term involving $\text{li}(t)$ $li (t)$ and a remainder term involving $f(t)$ $f (t)$ .
- The main term integral $\int_x^\infty t^{-s} d\text{li}(t)$ is transformed via substitution ( $u = \log t$ ) into the Exponential Integral function, $E_1(z)$ .
- The remainder term is analyzed to show that it vanishes or becomes negligible in the limit as $x \to \infty$ for $\Re(s) > \frac{1}{2}$ .

3. Key Contributions

The paper derives a new series expansion and limit formula for $P(s)$ :

Theorem 1:
$P(s) = \lim_{x \to \infty} \left( \sum_{p \le x} \frac{1}{p^s} + E_1[(s-1)\log(x)] \right)$

Domain of Validity: This expression is analytic for $\Re(s) > \frac{1}{2}$ .
Branch Cut: The formula requires a branch cut on the interval $(\frac{1}{2}, 1]$ due to the logarithmic nature of the exponential integral $E_1(z)$ .
Definition of $E_1(z)$ : The exponential integral is defined as $E_1(z) = \int_z^\infty \frac{e^{-t}}{t} dt$ , with a series expansion involving the Euler-Mascheroni constant $\gamma$ and logarithmic terms.
Simplification: The author demonstrates that the boundary terms arising from the integration by parts (specifically involving $\pi(x) - \text{li}(x)$ ) do not improve convergence and can be dropped in the limit, simplifying the final expression to the sum of the partial prime series and the exponential integral correction.

4. Results and Verification

Numerical Verification: The author implemented the derived formula (Equation 3) and the standard Möbius inversion formula (Equation 2) using Pari/GP.
Comparative Plots:
- Real Domain: Plots comparing the real part of $P(s)$ for $s \in [0.5001, 2]$ show a "near perfect reproduction" between the two methods. Deviations are noted only very close to $s=0.5$ due to convergence issues.
- Complex Domain: Plots along the vertical line $\Re(s) = 0.75$ (for imaginary parts $t \in [0.1, 50]$ ) demonstrate a perfect match for both the real and imaginary components of $P(s)$ .
Code Availability: The paper includes a Pari/GP script (Listing 1) that defines the functions and generates the verification plots.

5. Significance

Extension of Domain: This work successfully extends the analytic continuation of the Prime Zeta Function from the known boundary of $\Re(s) > 0$ down to $\Re(s) > \frac{1}{2}$ , contingent on the Riemann Hypothesis.
Computational Utility: The derived formula provides a computationally efficient method to evaluate $P(s)$ in the critical strip without relying on the Möbius inversion formula, which involves summing over all integers $n$ and requires evaluating $\zeta(ns)$ (which can be computationally expensive or unstable near poles).
Theoretical Insight: The result highlights the deep connection between the distribution of primes (via $\pi(x)$ and RH), the logarithmic integral, and the exponential integral, offering a new perspective on the singularities of $P(s)$ .

In summary, Kawalec provides a rigorous derivation and numerical validation of a simplified limit formula that allows the Prime Zeta Function to be evaluated in the half-plane $\Re(s) > 1/2$ , bridging a significant gap in the analytic theory of prime distributions under the assumption of the Riemann Hypothesis.