Multi-Scale Moment Inequalities and the Architecture of the Critical Line

Introduction

The Riemann Hypothesis remains one of the most profound challenges in mathematics, asserting that all non-trivial zeros of the Riemann zeta function lie on the critical line where the real part of s equals 1/2. Modern research increasingly relies on the behavior of Dirichlet polynomials to approximate the zeta function. A significant advancement in this field is presented in arXiv:1005.3932, which develops sharp moment and maximal inequalities for families of trigonometric polynomials indexed by time and truncation parameters.

Mathematical Framework

The core of the approach involves analyzing the 2q-th moments of Dirichlet polynomials. These polynomials serve as truncated approximations of the zeta function on the critical line. The research identifies that the fluctuations of these sums are governed by the arithmetic properties of primes and can be modeled as weakly correlated variables.

Moment Bounds and Gaussian Scaling

The study establishes that high moments of Dirichlet polynomials can be bounded by a factorial term (q!) multiplied by the q-th power of the L2 norm. This Gaussian-like behavior is modified by an arithmetic spacing parameter, xi, which measures the separation of frequencies. This framework allows for rigorous control over large deviations of the zeta function, which is a prerequisite for establishing zero-free regions.

Increment and Maximal Inequalities

A central technical device in arXiv:1005.3932 is an increment bound that relates the difference between polynomial values at two different times to a coefficient-weighted pseudo-metric. Furthermore, the paper provides uniform bounds on the suprema of these polynomials over ranges of truncation. This multiscale control is essential for managing the sensitive fluctuations of the Euler product near the critical line.

Novel Research Pathways

1. Zero-Density Improvements via Moment Optimization

By optimizing the choice of moments and interval lengths, researchers can derive contradictions if too many zeros are assumed to lie off the critical line. Utilizing the supremum bounds from the paper, one could potentially establish new zero-density theorems that significantly improve upon current classical bounds. The methodology involves applying the moment bounds with an optimally chosen order q related to the logarithm of the height T.

2. The Lindelof Hypothesis and Supremum Norms

The Lindelof Hypothesis, a known consequence of the Riemann Hypothesis, suggests that the growth of the zeta function on the critical line is slower than any power of T. The chaining arguments and maximal inequalities from the source paper provide a concrete mechanism to tackle this by bounding the supremum norm of Dirichlet polynomials over short intervals. This approach leverages the q-th moment bounds to apply the Borell-TIS inequality to the approximating stochastic process.

3. Spectral Repulsion and Frequency Spacing

There is a hypothesized duality between the spacing of prime logarithms and the repulsion of zeta zeros. The off-diagonal terms in the moment expansions analyzed in the paper relate to the small denominator problem. Establishing a formal link between the xi-spacing of prime frequencies and the Gaussian Unitary Ensemble statistics of zeros could lead to a proof of Montgomery’s pair correlation conjecture.

Wolfram Language Implementation

The following code allows for the numerical verification of truncation stability and the comparison of prime Dirichlet polynomials with the actual zeta function behavior on the critical line.

ClearAll["Global`*"]; t0 = 1000.0; tWindow = 50.0; primeListUpTo[n_] := Prime[Range[PrimePi[n]]]; SPrimeSmooth[N_, t_] := Module[{ps, logs}, ps = primeListUpTo[N]; logs = Log[ps]; Sum[(ps[[k]]^(-1/2) * Exp[-ps[[k]]/N]) * Exp[-I * t * logs[[k]]], {k, 1, Length[ps]}] ]; ZCrit[t_] := Zeta[1/2 + I t]; LogAbsZCrit[t_] := Log[Abs[ZCrit[t]]]; Plot3D[Re[SPrimeSmooth[N, t]], {t, t0 - tWindow, t0 + tWindow}, {N, 200, 2000}, PlotPoints -> 40, Mesh -> None, ColorFunction -> "ThermometerColors", AxesLabel -> {"t", "N", "Re S_N(t)"}]

Conclusions

The insights from arXiv:1005.3932 provide a robust toolkit for treating Dirichlet polynomials as weakly correlated random variables. The factorial growth of moments and the multiscale stability across truncations offer a structural blueprint for proving that zeta function values do not cluster in ways that would violate the Riemann Hypothesis. Future work should focus on optimizing the spacing parameters for prime frequencies and integrating these maximal inequalities with existing mollifier machinery.