Dirichlet-type representations of the Riemann zeta function take the form
F(σ,t) = ∑_(n=1)^∞ n^(-σ) e^(-itlogn), s=σ+it, 0<σ<1
where zeros correspond to exact cancellation among weighted oscillatory terms. This work develops a structural framework for analyzing such cancellation by studying the interaction between magnitude decay and phase alignment.
Within this framework, the vanishing of the sum is shown to force the emergence of localized cancellation blocks, for which diffuse phase alignm…
Read moreDirichlet-type representations of the Riemann zeta function take the form
F(σ,t) = ∑_(n=1)^∞ n^(-σ) e^(-itlogn), s=σ+it, 0<σ<1
where zeros correspond to exact cancellation among weighted oscillatory terms. This work develops a structural framework for analyzing such cancellation by studying the interaction between magnitude decay and phase alignment.
Within this framework, the vanishing of the sum is shown to force the emergence of localized cancellation blocks, for which diffuse phase alignment is quantitatively impossible. As a consequence, cancellation necessarily concentrates into interacting configurations satisfying density, phase, and magnitude constraints. These constraints are not imposed but arise as necessary structural conditions for cancellation.
Assuming the presence of a zero with σ∈(1/2,1), the vanishing of the first Fourier coefficient of the associated weighted phase distribution induces a global symmetry condition. In combination with the decay of the weights n^(-σ), this symmetry forces exponential localization of dominant contributions and constrains them to an approximately linearly spaced subsequence in logn.
Substitution of this structure reduces the sum to a weighted alternating exponential model with perturbations arising from integer constraints. The corresponding ideal alternating system is shown to remain strictly non-vanishing, and the perturbative contributions are bounded in such a way that they cannot produce cancellation of the required magnitude.
The result is a structural obstruction: any admissible cancellation configuration within this oscillatory Dirichlet framework must satisfy a rigid system of constraints that is incompatible for σ≠1/2. At the critical line σ=1/2, the decay becomes scale-neutral, and the localization mechanism fails, removing the obstruction.
This analysis demonstrates how structured constraint frameworks can force rigidity in oscillatory systems and collapse admissible cancellation configurations, providing a systematic method for studying phase–magnitude interactions in Dirichlet-type sums.
