[Paper Review] Notes on the Zeros of Riemann's Zeta Function

Michael Milgram|arXiv (Cornell University)|Nov 6, 2009

Analytic Number Theory Research12 references3 citations

TL;DR

This paper proposes a novel analysis of the Riemann zeta function’s functional equation by decomposing it into real and imaginary components, showing that non-trivial full-zeros can only occur on the critical line σ = 1/2. It derives two transcendental equations whose solutions locate all zeros on this line, though subsequent corrections invalidate the claim that this proves the Riemann Hypothesis.

ABSTRACT

The functional equation for Riemann's Zeta function is studied, from which it is shown why all of the non-trivial, full-zeros of the Zeta function $ζ(s)$ will only occur on the critical line {$σ=1/2$} where {$s=σ+I ρ$}, thereby establishing the truth of Riemann's hypothesis. Further, two relatively simple transcendental equations are obtained; the numerical solution of these equations locates all of the zeros of {$ζ(s)$} on the critical line.

Motivation & Objective

• To investigate the location of non-trivial zeros of the Riemann zeta function using the functional equation in real and imaginary components.
• To determine whether the functional equation constrains full-zeros to the critical line σ = 1/2.
• To derive explicit transcendental equations whose solutions correspond to all zeros of ζ(s) on the critical line.
• To provide a numerical framework for locating both half-zeros and full-zeros of ζ(s) using real-variable analysis.

Proposed method

• Express the functional equation of ζ(s) in terms of real and imaginary parts, defining ζ_R(σ,ρ) and ζ_I(σ,ρ) as the real and imaginary components of ζ(σ + iρ).
• Introduce transformed functions ˜ζ_R and ˜ζ_I that represent the functional equation’s action on these components, forming a coupled system of equations.
• Apply a symmetry constraint requiring ˜ζ_R(σ,ρ_p) = ζ_R(σ,ρ_p) and ˜ζ_I(σ,ρ_p) = ζ_I(σ,ρ_p), which identifies candidate lines for zero locations.
• Define auxiliary functions N, D_R, and D_I to detect half-zeros (where only one component vanishes), with N² = 0 identifying all half-zeros on the critical line.
• Derive a transcendental equation (4.7): D_R ⋅ ζ'_I + N ⋅ ζ'_R = 0, which serves as a necessary condition for locating full-zeros when m = 1.
• Use asymptotic approximations (e.g., Stirling’s formula) to simplify expressions involving the gamma function ratio Γ_I/Γ_R for large ρ, enabling approximate forms of the equations.

Experimental results

Research questions

• RQ1 Can the functional equation of ζ(s) be used to constrain the location of non-trivial full-zeros to the critical line σ = 1/2?
• RQ2 Are there explicit transcendental equations whose solutions correspond exactly to the zeros of ζ(s) on the critical line?
• RQ3 Can the real and imaginary parts of ζ(s) be used to derive a numerical method for locating both half-zeros and full-zeros?
• RQ4 Does the symmetry implied by the functional equation force full-zeros to appear in symmetric pairs across σ = 1/2, and can this be used to restrict their location?
• RQ5 Can the ratio of derivatives ζ'_R / ζ'_I be used to isolate full-zeros via a solvable transcendental equation?

Key findings

• The functional equation in real and imaginary components implies that any full-zero of ζ(s) in σ ≤ 1/2 must be mirrored by a corresponding full-zero in σ ≥ 1/2, suggesting symmetry about σ = 1/2.
• The analysis identifies the critical line σ = 1/2 as the only possible region where non-trivial full-zeros of ζ(s) can exist, based on the coupling constraints.
• Equation (4.1): C_m ⋅ cos(ρ_π) − C_p ⋅ sin(ρ_π) = 0, is shown to locate all half-zeros of ζ(s) on the critical line, with solutions alternating between ζ_R = 0 and ζ_I = 0.
• Equation (4.7): D_R ⋅ ζ'_I + N ⋅ ζ'_R = 0, serves as a necessary condition for locating full-zeros of ζ(s) on the critical line, though it is not sufficient and requires derivative information.
• The paper claims that the only region compatible with the functional equation’s constraints is σ = 1/2, leading to the conclusion that the Riemann Hypothesis is true—though this claim is later retracted.
• Subsequent corrections (2015) invalidate the original claim of proving the Riemann Hypothesis, citing a counterexample in a follow-up paper, though the analysis of ζ(1/2 + iρ) remains valid.