Analytical continuation of Euler prime product for $\Re(s)>\tfrac{1}{2}$ assuming (RH)

Assuming the Riemann Hypothesis, this paper introduces a new factor to analytically continue the Euler prime product to the region $\Re(s) > \tfrac{1}{2}$ (excluding the pole at $s=1$), discusses the recovery of Mertens's 3rd Theorem, and provides a Pari/GP script for numerical verification.

The Gist

The Big Picture: Fixing a Broken Bridge

Imagine the Riemann Zeta function ($\zeta(s)$) as a massive, magical bridge that connects the world of simple numbers to the mysterious world of prime numbers (2, 3, 5, 7...).

For a long time, mathematicians knew how to cross this bridge, but only on one side. Specifically, they could only walk across it safely if they stayed in the "Safe Zone" where the real part of the number $s$ is greater than 1.

The bridge is built using an Euler Product. Think of this product as a chain made of links, where every link is a prime number.

• The Chain: $\prod (1 - 1/p^s)^{-1}$
• The Problem: If you try to walk into the "Danger Zone" (where the real part of $s$ is between 0.5 and 1), the chain starts to unravel. The links get too heavy, and the math breaks down. The bridge collapses.

The Goal of this Paper:
The author, Artur Kawalec, wants to extend the bridge so we can walk all the way down to the "Critical Line" (where the real part is 0.5), assuming a famous hypothesis called the Riemann Hypothesis (RH) is true. He wants to fix the bridge so it doesn't collapse in the Danger Zone.

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The Solution: Adding a "Stabilizer"

How does he fix the bridge? He realizes that the chain of prime numbers (the Euler Product) is missing a crucial piece of support when we get close to the edge.

The Analogy: The Leaning Tower
Imagine the Euler Product is a tower of blocks.

• On the safe side ($s > 1$), the tower stands perfectly straight.
• As you move toward the edge ($0.5 < s < 1$), the tower starts to lean dangerously. It's about to fall.

Kawalec says, "We don't need to rebuild the tower; we just need to add a counterweight."

He introduces a new mathematical factor: $e^{E_1[(s-1)\log x]}$.

• What is this? It's a special "correction factor" (a type of exponential integral).
• What does it do? It acts like a heavy anchor or a counterweight. When you attach this anchor to the leaning tower (the partial product of primes), it pulls the tower back into balance.

The Result:
With this new anchor attached, the bridge becomes stable. You can now walk from the Safe Zone all the way down to the Critical Line ($s = 0.5$), provided the Riemann Hypothesis is true.

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The "Magic Trick" at the Edge (s = 1)

There is a specific spot on the bridge, right at $s=1$, where the bridge is supposed to have a giant hole (a "pole"). This is where the famous Mertens' Theorem lives.

Usually, when you try to calculate the product of primes at this spot, the numbers go to infinity. It's like trying to fill a bucket with a hole in the bottom.

Kawalec shows that his new "anchor" formula is so smart that it knows exactly how to behave at this hole.

• If you look at the formula near the hole, the "anchor" cancels out the infinity perfectly.
• When you do the math, it magically recreates the famous rule that describes how prime numbers grow (Mertens' 3rd Theorem).
• The Takeaway: The new formula isn't just a patch; it's a perfect fit that respects the old rules even at the most dangerous spot.

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Does it work for other bridges?

Yes! The author shows that this "anchor" technique isn't just for the main Zeta bridge.

• He applies it to the Reciprocal Zeta (a bridge that goes the other way).
• He applies it to other complex combinations of prime products.

In every case, adding this specific "exponential integral" correction factor allows the math to work in the Danger Zone where it previously failed.

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The Proof: The Computer Simulation

Math is great, but you have to prove it works in the real world. Kawalec wrote a computer script (using a program called Pari/GP) to test his theory.

The Experiment:

1. He built a "partial bridge" using the first 1,000, 10,000, and 100,000 prime numbers.
2. He added his "anchor" to these partial bridges.
3. He compared the result to the actual, known Riemann Zeta function.

The Findings:

• At first glance: The two lines (the real function and his new formula) matched perfectly.
• Near the edge (0.5): When he got very close to the critical line, the lines started to wiggle or oscillate a little bit.
• The Fix: He realized this was just because he hadn't used enough primes yet. When he increased the number of primes in the simulation (making the bridge longer), the wiggles smoothed out, and the lines matched again.

Conclusion: The more primes you use, the more perfect the match becomes. This suggests the formula is correct.

Summary in One Sentence

The author discovered a clever mathematical "counterweight" that allows us to extend the famous formula for prime numbers into a dangerous, previously unreachable zone, assuming the Riemann Hypothesis is true, and computer tests confirm it works beautifully.

Technical Summary

1. Problem Statement

The Riemann zeta function, $\zeta(s)$, is classically defined by the Euler product over primes:
$$\zeta(s) = \prod_{p} \left(1 - \frac{1}{p^s}\right)^{-1}$$
This product is absolutely convergent only for $\Re(s) > 1$. In the critical strip $1/2 < \Re(s) \leq 1$, the Euler product diverges, preventing the direct use of the product form to analyze the zeta function in this region.

The paper addresses the challenge of analytically continuing the Euler prime product from the domain $\Re(s) > 1$ down to the domain $\Re(s) > 1/2$ (excluding the pole at $s=1$), assuming the truth of the Riemann Hypothesis (RH).

2. Methodology

The author's approach relies on analyzing the logarithmic expansion of the Euler product and isolating the component responsible for the convergence limit.

• Logarithmic Decomposition:
  By taking the logarithm of the Euler product, the author expands it into a series involving the Prime Zeta Function, $P(s) = \sum_p p^{-s}$:
  $$\log \zeta(s) = \sum_{n=1}^{\infty} \frac{P(ns)}{n} = P(s) + \sum_{n=2}^{\infty} \frac{P(ns)}{n}$$

  • The second term ($\sum_{n=2}^{\infty}$) converges for $\Re(s) > 1/2$.
  • The first term, $P(s)$, is the bottleneck; it only converges for $\Re(s) > 1$.

• Asymptotic Correction:
  The paper utilizes a previously established asymptotic expansion for the partial sum of the Prime Zeta Function. As $x \to \infty$:
  $$\sum_{p \leq x} \frac{1}{p^s} = P(s) - E_1[(s-1)\log x] + O\left(x^{1/2-s} \log x\right)$$
  Here, $E_1(z)$ is the Exponential Integral defined as $\int_z^\infty \frac{e^{-t}}{t} dt$. Under the assumption of RH, this expansion holds for $\Re(s) > 1/2$.

• The Correction Factor:
  To recover the full $P(s)$ term from the partial sum (which is what the finite Euler product calculates), the author introduces a compensating factor. By rearranging the asymptotic formula, the missing $P(s)$ component is recovered by adding the term $E_1[(s-1)\log x]$.

3. Key Contributions

The paper presents a novel formula that extends the validity of the Euler product representation of $\zeta(s)$.

• Theorem 1 (Main Result):
  Assuming the Riemann Hypothesis, the Riemann zeta function for $\Re(s) > 1/2$ (where $s \neq 1$) can be expressed as:
  $$\zeta(s) = \lim_{x \to \infty} e^{E_1[(s-1)\log x]} \prod_{p \leq x} \left(1 - \frac{1}{p^s}\right)^{-1}$$
  This formula effectively "patches" the divergence of the standard Euler product in the critical strip by multiplying the partial product by an exponential integral term that accounts for the tail of the prime sum.

• Generalization to Other Products:
  The technique is shown to be applicable to other Euler products, specifically:

  1. The Reciprocal Zeta Function: $\frac{1}{\zeta(s)} = \lim_{x \to \infty} e^{-E_1[(s-1)\log x]} \prod_{p \leq x} (1 - p^{-s})$.
  2. The Ratio $\frac{\zeta(2s)}{\zeta(s)}$: $\frac{\zeta(2s)}{\zeta(s)} = \lim_{x \to \infty} e^{-E_1[(s-1)\log x]} \prod_{p \leq x} (1 + p^{-s})^{-1}$.
     In both cases, the correction factor allows convergence down to $\Re(s) > 1/2$.

• Recovery of Mertens' Theorem:
  The paper demonstrates that taking the limit $s \to 1$ in the new formula naturally recovers Mertens' 3rd Theorem:
  $$\prod_{p \leq x} \left(1 - \frac{1}{p}\right)^{-1} \sim e^\gamma \log x$$
  This is achieved by utilizing the series expansion of $E_1(z)$ near zero, which introduces the Euler-Mascheroni constant $\gamma$ and the logarithmic divergence required to match the pole of $\zeta(s)$ at $s=1$.

4. Results and Numerical Verification

The author provides numerical evidence using Pari/GP to validate the theoretical findings.

• Visual Comparison:
  • Fig 1: Plots $\zeta(s)$ against the analytically continued Euler product for $x=10^6$ in the range $0.5 < \Re(s) < 2$. The curves show a "perfect match," with minor deviations only appearing very close to the critical line $\Re(s) = 0.5$.
  • Figs 2 & 3: Plots the modulus of the function along vertical lines ($\sigma = 0.8$ and $\sigma = 0.55$) for $t \in [0, 50]$.
    • At $\sigma = 0.8$, the match is near perfect.
    • At $\sigma = 0.55$ (closer to the critical line), oscillations appear in the finite product approximation.
  • Fig 4: Increasing the limit $x$ from $10^3$ to $10^5$ at $\sigma = 0.55$ smooths out these oscillations, confirming that the error term $O(x^{1/2-s} \log x)$ diminishes as $x \to \infty$.

5. Significance

• Theoretical Insight: The paper provides a concrete mechanism to understand why the Euler product fails for $\Re(s) \leq 1$ and offers a precise correction term derived from the asymptotic behavior of the Prime Zeta Function.
• Conditional Validity: The results are contingent on the Riemann Hypothesis. If RH is true, the Prime Zeta Function has a specific analytic structure that allows this continuation; if RH is false, the error terms would behave differently, and this specific continuation might not hold.
• Computational Utility: The derived formula offers a potential method for computing values of $\zeta(s)$ in the critical strip using prime products, provided the exponential integral correction is applied. This bridges the gap between the multiplicative structure of primes and the analytic properties of the zeta function in the critical strip.
• Unification: It unifies the behavior of the zeta function at the pole ($s=1$) with its behavior in the critical strip, showing that the divergence at $s=1$ (Mertens' theorem) and the convergence issues in the strip are governed by the same exponential integral term.