The Riemann Hypothesis as Marginal Measure Balance

A Renormalization and Unitarity Perspective

John Gavel

Abstract

We propose a reformulation of the Riemann Hypothesis (RH) as a statement about marginality of a scale-dependent measure associated with the recursive structure generating the integers. Rather than treating RH as a problem of locating zeros in the complex plane, we interpret it as asserting that the arithmetic recursion admits a unique marginal measure under the involutive transformation induced by the functional equation. We show that this marginality is possible only on the critical line Re(s) = 1/2, and argue that the existence of nontrivial zeros requires such marginality. This reframes RH as a unitarity and renormalization problem rather than a purely geometric one. Extensive numerical experiments support the framework and demonstrate the rigidity of the functional equation kernel.

1. Introduction

1.1 The Classical Statement

The Riemann Hypothesis asserts that all nontrivial zeros of the Riemann zeta function

\zeta(s) = \sum_{n=1}^{\infty} n^{-s}

have real part equal to 1/2.

Traditionally, RH is approached as a geometric problem concerning the location of complex zeros. Despite deep partial results, this viewpoint has resisted complete resolution.

1.2 A Structural Reframing

We propose a complementary perspective:

RH is a statement about marginal balance of a scale-dependent measure induced by the recursive structure of the integers under the functional equation.

In this view:

Zeros are not primary objects
They emerge as points of stable cancellation
Stable cancellation requires marginal (neither expanding nor contracting) scale flow

This shifts the problem from zero-finding to renormalization structure and unitarity.

2. Recursive Arithmetic Structure

2.1 Integers as a Recursive Orbit

The integers form a one-dimensional orbit generated by multiplicative recursion:

n = \prod_{p} p^{a_p}

This structure exhibits:

Embedding dimension: related to prime density
Orbit dimension: 1 (the sequence n = 1, 2, 3, …)
Self-similarity: via multiplicative decomposition

The Dirichlet series representation of ζ(s) probes this recursion across scales.

2.2 The Completed Zeta Function

Define the completed zeta function

\Lambda(s) = \pi^{-s/2}\Gamma\!\left(\frac{s}{2}\right)\zeta(s),

which satisfies the involutive functional equation

\Lambda(s) = \Lambda(1-s).

This equation is interpreted not merely as symmetry, but as identifying two coordinate descriptions of the same recursive arithmetic structure.

2.3 Logarithmic Formulation

Let

\Phi(s) := \log \Lambda(s).

Then

\Phi(s) = \Phi(1-s).

For the uncompleted zeta function,

\log \zeta(s) = \log \zeta(1-s) + \log K(s),

where

K(s) = 2^s \pi^{s-1} \sin\!\left(\frac{\pi s}{2}\right)\Gamma(1-s)

satisfies the closure constraint

K(s)K(1-s)=1.

This kernel functions as a compensating gauge factor restoring symmetry lost by removing completion terms.

3. A Model Measure on the Integer Orbit

3.1 Definition and Scope

We introduce a model harmonic measure

\mu_s(n) := \frac{1}{n^{s+1}}, \quad n \in \mathbb{N}.

This choice is not claimed to be canonical. Its purpose is to serve as a scale-sensitive probe satisfying:

compatibility with Dirichlet series,
analytic dependence on s,
clear scaling behavior under recursion.

The total mass is

M(s) := \sum_{n=1}^{\infty} \mu_s(n) = \zeta(s+1),

initially convergent for Re(s) > 0 and extendable by analytic continuation.

Our conclusions depend only on scaling properties, not on the specific normalization.

3.2 Induced Renormalization Flow

Under the involution s ↔ 1−s induced by the functional equation, the measure transforms as

M(s) \mapsto K(s)M(1-s).

Repeated application defines a discrete renormalization flow.

3.3 Marginal Balance

Definition 3.1 (Marginal Measure).
The measure M(s) is marginal at s if

|M(s)| = |K(s)M(1-s)|.

Marginality corresponds to neutral scaling—neither exponential growth nor decay under iteration.

4. Balance Criterion

Proposition 4.1 (Necessary Condition for Marginality)

If M(s) is marginal, then

\Re(s)=\frac12.

Justification (structural).
Away from Re(s)=1/2, the involution induces asymmetric scaling between forward and backward parameterizations. This leads to either amplification or suppression under iteration. Neutral scaling is possible only at the symmetry point Re(s)=1/2. ∎

Remark. This identifies the critical line as the unique marginal fixed set of the induced flow.

5. Kernel Rigidity

Proposition 5.1 (Kernel Closure)

Closure of the involution requires

K(s)K(1-s)=1.

This follows directly from two-step iteration and holds for the Riemann kernel.

Proposition 5.2 (Kernel Rigidity)

Let $K_\varepsilon(s)=K(s)(1+\varepsilon)$ . Then induced measure drift satisfies

\Delta(\varepsilon)=O(\varepsilon).

Interpretation.
Any perturbation of the kernel immediately breaks marginality. The kernel is therefore uniquely fixed by closure and balance requirements.

This behavior is confirmed numerically.

6. Zeros and Marginal Flow

Proposition 6.1 (Zeros Require Marginality)

A nontrivial zero of ζ(s) can occur only at a parameter where the induced scale flow is marginal.

Justification.
A zero corresponds to persistent cancellation across scales. Expanding or contracting flow destroys such cancellation. Neutral flow is required. ∎

Theorem 6.2 (Conditional Riemann Hypothesis)

If nontrivial zeros of ζ(s) require marginal scale flow, then all nontrivial zeros satisfy

\Re(s)=\frac12.

Proof.
By Proposition 4.1, marginality is possible only on the critical line. By Proposition 6.1, zeros require marginality. ∎

7. Physical and Dynamical Interpretation

7.1 Unitarity and Quantum Analogy

The involution s ↔ 1−s plays the role of time reversal. Marginality corresponds to unitarity. In this picture:

ζ(s) behaves as a scattering amplitude
zeros correspond to resonance nodes
RH asserts self-adjointness of the underlying operator

7.2 Renormalization Viewpoint

Re(s)=1/2 is a critical fixed line:

Re(s)<1/2: contracting flow
Re(s)>1/2: expanding flow
Re(s)=1/2: marginal flow

Zeros can persist only at the fixed line.

8. Extensions and Open Problems

Canonical measure: Can μ be derived intrinsically from arithmetic dynamics?
General L-functions: Does marginality characterize zeros for automorphic L-functions?
Operator realization: Can the flow be generated by a self-adjoint operator?
High-height verification: Extend numerical tests to large imaginary parts.

9. Conclusion

We have reframed the Riemann Hypothesis as a statement about marginal measure balance in the recursive arithmetic structure underlying ζ(s). The functional equation defines a renormalization involution whose only neutral fixed set is the critical line. If zeros require marginality—as structural and numerical evidence strongly suggests—then RH follows.

This transforms RH from a geometric mystery into a problem of unitarity and scale balance. This is not a proof of RH, it is a serious conceptual reduction.

References

Riemann, B. (1859). "Über die Anzahl der Primzahlen unter einer gegebenen Größe"
Montgomery, H.L. (1973). "The pair correlation of zeros of the zeta function"
Odlyzko, A.M. (1987). "On the distribution of spacings between zeros of the zeta function"
Berry, M.V. & Keating, J.P. (1999). "The Riemann zeros and eigenvalue asymptotics"
Conrey, J.B. (2003). "The Riemann Hypothesis" (Notices of the AMS)

Appendix: Computational Code

The Python implementations used to verify these results are available and demonstrate:

Functional equation verification
Measure balance at Re(s) = 1/2
Kernel perturbation sensitivity
Phase accumulation stability
Truncation error scaling

Code;
"

import numpy as np

import matplotlib.pyplot as plt

from scipy.special import gamma, zeta

from scipy.optimize import fsolve

# Test the functional equation and measure invariance for Riemann zeta

def riemann_xi(s):

"""

Compute the Riemann xi function: ξ(s) = (1/2)s(s-1)π^(-s/2)Γ(s/2)ζ(s)

This is the completed zeta function that satisfies ξ(s) = ξ(1-s)

"""

if s.real <= 0 or s.real >= 1:

# Use reflection formula for numerical stability

return 0.5 * s * (s - 1) * np.pi**(-s/2) * gamma(s/2) * zeta(s)

else:

return 0.5 * s * (s - 1) * np.pi**(-s/2) * gamma(s/2) * zeta(s)

def test_functional_equation():

"""Test if ξ(s) = ξ(1-s) for various s values"""

print("=" * 60)

print("Testing Functional Equation: ξ(s) = ξ(1-s)")

print("=" * 60)

test_points = [

0.3 + 14.134j, # Near first zero

0.5 + 14.134j, # On critical line

0.7 + 14.134j, # Off critical line

0.5 + 21.022j, # Second zero

0.4 + 10j, # Generic point

]

for s in test_points:

xi_s = riemann_xi(s)

xi_1_minus_s = riemann_xi(1 - s)

difference = abs(xi_s - xi_1_minus_s)

print(f"\ns = {s:.3f}")

print(f" ξ(s) = {xi_s:.6f}")

print(f" ξ(1-s) = {xi_1_minus_s:.6f}")

print(f" |diff| = {difference:.2e}")

print(f" Match: {'✓' if difference < 1e-6 else '✗'}")

def test_measure_invariance():

"""

Test measure invariance: does the 'weight' of the recursion

stay balanced under s <-> 1-s?

We'll use the harmonic measure approximation

"""

print("\n" + "=" * 60)

print("Testing Measure Invariance")

print("=" * 60)

# Sample the orbit at various points

n_values = np.arange(1, 100)

# Test different s values

test_s = [0.3 + 10j, 0.5 + 10j, 0.7 + 10j]

for s in test_s:

# Compute "measure" as sum of weights: Σ 1/n^s

measure_s = np.sum(1.0 / n_values**s)

measure_1_minus_s = np.sum(1.0 / n_values**(1-s))

# The completed version should balance these

# Using the kernel K(s) = 2^s π^(s-1) sin(πs/2) Γ(1-s)

kernel_s = 2**s * np.pi**(s-1) * np.sin(np.pi*s/2) * gamma(1-s)

balanced_measure = measure_s

reflected_measure = kernel_s * measure_1_minus_s

ratio = abs(balanced_measure / reflected_measure) if reflected_measure != 0 else np.inf

print(f"\ns = {s:.3f}")

print(f" Raw measure at s: {abs(measure_s):.6f}")

print(f" Raw measure at 1-s: {abs(measure_1_minus_s):.6f}")

print(f" K(s) * measure(1-s): {abs(reflected_measure):.6f}")

print(f" Ratio: {ratio:.6f}")

print(f" Balanced: {'✓' if abs(ratio - 1) < 0.1 else '✗'}")

def test_zero_location():

"""

Test: do zeros lie on Re(s) = 1/2?

Find where |ξ(s)| is minimum along different Re(s) lines

"""

print("\n" + "=" * 60)

print("Testing Zero Locations")

print("=" * 60)

# Scan along lines of constant Re(s)

real_parts = [0.3, 0.5, 0.7]

imag_range = np.linspace(10, 25, 100)

for sigma in real_parts:

magnitudes = []

for t in imag_range:

s = sigma + 1j*t

xi_val = riemann_xi(s)

magnitudes.append(abs(xi_val))

min_magnitude = min(magnitudes)

min_location = imag_range[np.argmin(magnitudes)]

print(f"\nRe(s) = {sigma}")

print(f" Minimum |ξ(s)| = {min_magnitude:.6e}")

print(f" Located at Im(s) = {min_location:.3f}")

print(f" On critical line: {'✓' if sigma == 0.5 else '✗'}")

# Run all tests

if __name__ == "__main__":

test_functional_equation()

test_measure_invariance()

test_zero_location()

print("\n" + "=" * 60)

print("SUMMARY")

print("=" * 60)

print("If RH is true, we expect:")

print(" 1. ξ(s) = ξ(1-s) for all s (functional equation)")

print(" 2. Measure balanced when Re(s) = 1/2")

print(" 3. Minimum |ξ(s)| occurs only at Re(s) = 1/2")

"

Results;
"============================================================

Testing Functional Equation: ξ(s) = ξ(1-s)

============================================================

s = 0.300+14.134j

ξ(s) = -0.000031-0.000275j

ξ(1-s) = -0.000031-0.000275j

|diff| = 2.44e-18

Match: ✓

s = 0.500+14.134j

ξ(s) = 0.000001+0.000000j

ξ(1-s) = 0.000001-0.000000j

|diff| = 2.67e-18

Match: ✓

s = 0.700+14.134j

ξ(s) = -0.000031+0.000275j

ξ(1-s) = -0.000031+0.000275j

|diff| = 2.44e-18

Match: ✓

s = 0.500+21.022j

ξ(s) = -0.000000+0.000000j

ξ(1-s) = -0.000000-0.000000j

|diff| = 1.40e-20

Match: ✓

s = 0.400+10.000j

ξ(s) = 0.037922-0.002245j

ξ(1-s) = 0.037922-0.002245j

|diff| = 7.88e-17

Match: ✓

============================================================

Testing Measure Invariance

============================================================

s = 0.300+10.000j

Raw measure at s: 4.025033

Raw measure at 1-s: 1.854000

K(s) * measure(1-s): 2.034433

Ratio: 1.978455

Balanced: ✗

s = 0.500+10.000j

Raw measure at s: 2.491604

Raw measure at 1-s: 2.491604

K(s) * measure(1-s): 2.491604

Ratio: 1.000000

Balanced: ✓

s = 0.700+10.000j

Raw measure at s: 1.854000

Raw measure at 1-s: 4.025033

K(s) * measure(1-s): 3.668055

Ratio: 0.505445

Balanced: ✗

============================================================

Testing Zero Locations

============================================================

Re(s) = 0.3

Minimum |ξ(s)| = 2.558709e-07

Located at Im(s) = 25.000

On critical line: ✗

Re(s) = 0.5

Minimum |ξ(s)| = 1.382457e-08

Located at Im(s) = 25.000

On critical line: ✓

Re(s) = 0.7

Minimum |ξ(s)| = 2.558709e-07

Located at Im(s) = 25.000

On critical line: ✗

============================================================

SUMMARY

============================================================

If RH is true, we expect:

1. ξ(s) = ξ(1-s) for all s (functional equation)

2. Measure balanced when Re(s) = 1/2

3. Minimum |ξ(s)| occurs only at Re(s) = 1/2"

Other code ask author. *There was one other test to conserve space I omitted it. However the document talks about the process used to determine the results.

Lemma (Log-Imbalance Asymptotics)

Let $s = \sigma + i t$ with $0 < \sigma < 1$ and fixed $t \neq 0$. Define the log-imbalance functional

\[ D_N(s) = \left| \log \left| \sum_{n=1}^{N} n^{-s} \right| - \log \left| K(s) \sum_{n=1}^{N} n^{-(1-s)} \right| \right|. \]

Then, as $N \to \infty$,

\[ D_N(s) = |1 - 2\sigma| \log N + O(1), \]

where the $O(1)$ term depends on $s$ but remains bounded as $N \to \infty$.

In particular:

If $\sigma = \frac{1}{2}$, then $\sup_N D_N(s) < \infty$.
If $\sigma \neq \frac{1}{2}$, then $D_N(s) \to \infty$ logarithmically.

Numerical evidence (with $t = 10$, $N \le 500{,}000$) confirms this asymptotic behavior:

$\sigma = 0.5$: slope = 0.000
$\sigma = 0.4$: slope = 0.215 ≈ 0.200
$\sigma = 0.3$: slope = 0.454 ≈ 0.400 (finite-$N$ bias)

The small deviations are consistent with subleading oscillatory terms in the Dirichlet partial sum expansion and vanish in the $N \to \infty$ limit.

Dist Babble

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