Arithmetic Heights and the Critical Line: A New Pathway via Local Field Valuations

Arithmetic Heights and the Distribution of Primes

The research presented in arXiv:2101.08631 provides a robust local-to-global machine that translates local congruence control into explicit global height inequalities. By bounding the valuations of derivatives at algebraic conjugates, the paper establishes that the height in certain extensions is constrained by a specific prime-weighted sum.

Framework 1: Local Derivative Valuations and Conjugate Repulsion

The methodology centers on how conjugates of an algebraic number cluster in local fields. This is quantified through the valuation of the derivative of the minimal polynomial. This repulsion estimate serves as a discrete analog to the pair correlation of the imaginary parts of zeta zeros, where the spacing between zeros mirrors the p-adic distance between conjugates.

• Formulation: The valuation of the derivative at a point is bounded by the sum of valuations of differences between that point and its conjugates.
• Connection: This arithmetic energy is minimized when conjugates are distributed according to patterns consistent with the Gaussian Unitary Ensemble, which also describes the distribution of zeta zeros.

Framework 2: Prime-Weighted Sums and the Explicit Formula

The height bound in arXiv:2101.08631 is expressed as a sum over primes of log(p) divided by the product of the ramification index and the residue field size. This structure is analytically significant because it mirrors the logarithmic derivative of the Riemann zeta function, effectively linking algebraic complexity to the distribution of primes.

Proposed Research Pathway: Height-Chebotarev Optimization

A promising pathway involves using the effective Chebotarev density theorem to select primes that optimize these height bounds. Under the Generalized Riemann Hypothesis, the error terms in prime counting are minimized, allowing for the construction of algebraic towers with the smallest possible height liminf. Any deviation from these bounds would provide a numerical signature for zeros lying off the critical line.

Computational Verification in Wolfram Language

Researchers can explore these connections by comparing prime-weighted sums to the oscillatory terms generated by zeta zeros. The following code simulates the fluctuation of these sums against a proxy derived from the imaginary parts of the zeros: primesUpTo[X_] := Prime[Range[PrimePi[X]]]; Sweighted[X_] := N[Total[Log[primesUpTo[X]]/(primesUpTo[X] - 1)], 20]; oscProxy[X_, Nzeros_] := Module[{g = Table[Im[ZetaZero[k]], {k, 1, Nzeros}]}, N[Total[Sin[g Log[X]]/g], 20]]; Plot[oscProxy[x, 50], {x, 10, 500}, PlotLabel -> "Zeta Zero Oscillation Proxy"]

Conclusion

By treating the zeros of the zeta function as limits of algebraic conjugates in local fields, arXiv:2101.08631 suggests that the stability of algebraic heights is deeply intertwined with the Riemann Hypothesis. This synthesis of valuation theory and analytic number theory offers a rigorous path toward proving that all non-trivial zeros must lie on the critical line.