[Paper Review] All the zeros of the Dirichlet eta function in the critical strip are on the critical line

James G. Andrews|arXiv (Cornell University)|Jan 10, 2012

Phase Equilibria and Thermodynamics1 references1 citations

TL;DR

This paper demonstrates that all non-trivial zeros of the Dirichlet eta function lie on the critical line ℜ(s) = 1/2, using fluid dynamics analogies from complex potential theory and established number-theoretic results. By modeling the eta function as an ideal fluid flow, it proves no zeros exist off the critical line in the critical strip, providing strong support for the Riemann hypothesis.

ABSTRACT

We describe the behaviour of the Dirichlet eta function in the critical strip, in terms of the potential flow of an ideal fluid. Using well-known results from complex potential theory and number theory, we show that the Dirichlet eta function has no zeros in the critical strip off the critical line, consistent with the Riemann hypothesis.

Motivation & Objective

• To investigate the distribution of non-trivial zeros of the Dirichlet eta function within the critical strip.
• To explore the connection between complex potential theory and the analytic behavior of L-functions.
• To provide a novel physical interpretation—via ideal fluid flow—of the zero-free region off the critical line.
• To establish a rigorous argument consistent with the Riemann hypothesis using known results from number theory and complex analysis.

Proposed method

• Modeling the Dirichlet eta function as a complex potential flow in the critical strip using conformal mapping techniques.
• Applying results from complex potential theory to analyze the flow's singularities, which correspond to the function's zeros.
• Using the functional equation of the eta function to relate its behavior across the critical line.
• Leveraging known theorems on the non-vanishing of certain zeta-related functions in the critical strip to constrain zero locations.
• Analyzing the streamlines and equipotential lines of the flow to infer the absence of zeros off ℜ(s) = 1/2.
• Combining number-theoretic properties of the eta function with topological constraints from fluid flow to exclude off-line zeros.

Experimental results

Research questions

• RQ1 Can the distribution of zeros of the Dirichlet eta function be analyzed using analogies from fluid dynamics?
• RQ2 Are there topological or geometric constraints in the complex potential flow that prevent zeros from existing off the critical line?
• RQ3 Does the behavior of the eta function as an ideal fluid flow support the absence of non-trivial zeros away from ℜ(s) = 1/2?
• RQ4 How do established results in complex potential theory and number theory jointly constrain the location of the eta function's zeros?
• RQ5 Can the Riemann hypothesis be approached through physical interpretations of L-functions via fluid flow models?

Key findings

• The Dirichlet eta function has no non-trivial zeros in the critical strip outside the critical line ℜ(s) = 1/2.
• The fluid flow analogy reveals that the absence of singularities in the flow field away from the critical line implies no corresponding zeros.
• The conformal structure of the eta function's complex potential ensures that any zero off the critical line would violate known analytic constraints.
• The proof relies on the consistency of the fluid flow model with established number-theoretic results, particularly those concerning the non-vanishing of related zeta functions.
• The absence of off-line zeros is topologically enforced by the flow's harmonic and analytic structure, aligning with the Riemann hypothesis.
• The argument is consistent with the broader conjecture that all non-trivial zeros of L-functions lie on the critical line.