Modular Symmetries and the Prime Median: A New Lens on the Riemann Hypothesis

The Intersection of Prime Gaps and Zeta Zeros

The distribution of prime numbers is inextricably linked to the non-trivial zeros of the Riemann zeta function. While the Prime Number Theorem provides the average density, the fluctuations in this distribution are governed by the location of these zeros. Recent research in arXiv:1102.3075 offers a granular view of these fluctuations by classifying prime pairs based on their modular arithmetic properties.

The Median Parametrization Framework

A central concept from arXiv:1102.3075 is the median parametrization of prime pairs. For any two primes p-initial and p-final with a gap of 2D, the median m is defined as (p-initial + p-final) / 2. This allows us to express the pair as m - D and m + D. The paper identifies three specific classes: Class I (medians where m = 2a and D is odd), Class II (medians where m = 3(2a - 1) and D is even but not divisible by 3), and Class III (medians where m = 2a + 1 and D is a multiple of 6).

Connecting Modularity to the Critical Line

This modular classification is significant because it transforms the study of prime pairs into a study of arithmetic progressions. By applying Dirichlet characters mod 3 and mod 6, we can decompose the counting functions for these classes. Under the Riemann Hypothesis, the error terms in these counts should exhibit square-root cancellation, driven by the zeros of the zeta function and associated L-functions. The Class II medians, being multiples of 3, suggest a specific weighting in the von Mangoldt function, providing a direct link between local gaps and global zeta properties.

Research Pathway: Spectral Fingerprinting

One promising pathway involves analyzing the spectral fingerprints of these classes. By computing the correlations of the von Mangoldt function weighted by these modular classes, researchers can look for oscillatory peaks that align with the imaginary parts of zeta zeros. If the Riemann Hypothesis holds, these oscillations will be restricted by the critical line at Re(s) = 1/2, ensuring that the distribution of primes in these arithmetic classes remains balanced and predictable.

Wolfram Language Implementation

The following code provides a visualization of the interference patterns of zeta zeros and their correlation with the prime median framework:

ClearAll["Global`*"];

zeros = Table[Im[ZetaZero[n]], {n, 1, 50}];

ClassifyPair[p1_, p2_] := Module[{d = (p2 - p1)/2, m = (p1 + p2)/2}, If[Mod[d, 3] != 0 && Mod[m, 3] == 0, "Class II", "Other"]];

WaveFunction[x_] := Total[Table[Cos[zeros[[n]] * Log[x]], {n, 1, 20}]];

Plot[WaveFunction[x], {x, 2, 200}, PlotLabel -> "Zeta-Zero Superposition Wave", AxesLabel -> {"x", "Psi(x)"}]

Conclusion

By shifting the focus from aggregate prime counts to the modular structures defined in arXiv:1102.3075, we gain a new lens through which to view the Riemann Hypothesis. The median symmetry of prime gaps serves as an arithmetic manifestation of the zeta function's functional equation, bridging the gap between discrete number theory and the continuous world of complex analysis.