Unifying Discrete Magma Algebra and Integral Kernels to Solve the Riemann Hypothesis

Introduction to the Magma-Integral Framework

The pursuit of the Riemann Hypothesis (RH) has traditionally relied on analytic number theory and spectral analysis. However, a recent mathematical development presented in arXiv:hal-04547463 introduces a transformative approach. This framework utilizes discrete associative magmas and complex integral representations to decompose the behavior of zeta-like functions into manageable algebraic and analytic components.

The core of this approach lies in the function f(s), where s = x + iy, which is represented as a weighted sum of oscillatory integrals. By mapping these kernels onto a discrete algebraic structure, researchers can study the symmetry of the critical line Re(s) = 1/2 through the lens of convolution rings.

Mathematical Frameworks and Discrete Structures

1. Convolution Rings on Discrete Magmas

The paper defines a magma (M, *) where the operation is discrete, meaning the set of factors for any element is finite. Functions over this magma form a ring under a specific convolution product. This algebraic structure generalizes Dirichlet convolution, which is fundamental to the study of multiplicative functions.

• Associativity: If the magma is associative, the resulting function ring is also associative. This allows for the construction of generalized zeta objects that maintain the structural properties of the Riemann zeta function.
• Algebraic Inversion: The presence of a convolution unit enables the derivation of Möbius-type inversion formulas within abstract factorization systems.

2. Integral Decomposition via Logarithmic Kernels

The function f(x + iy) is decomposed into two primary components, f0 and f1, using kernels g0 and g1. These components are expressed as integrals involving cosine and sine oscillations in a logarithmic variable u = ln(sqrt(1 + t^2)).

• Framework Formulation: The function is written as f(s) = w(s)f0(s) + i w(s)f1(s), where w(s) is a complex scaling factor.
• Oscillatory Dynamics: By using the Taylor series for trigonometric functions, the integrals are converted into moment series in powers of y. This allows for the analysis of zero-crossings by examining the signs of weighted moments.

Novel Research Pathways

Pathway A: The Kernel Symmetry Criterion

This approach investigates whether the ratio of the kernels g1 and g0 possesses an internal symmetry that forces the imaginary component of f(s) to vanish only when x = 1/2. By identifying the phase shift in the integral as x deviates from the critical line, one can establish bounds for the location of zeros.

Pathway B: Spectral Determinants of Log-Kernel Operators

By recasting the integral representations as Fourier transforms of measures on the logarithmic variable u, we can construct a self-adjoint operator T. If the zeros of the zeta function correspond to the eigenvalues of T, the Hilbert-Polya conjecture is realized, forcing all zeros onto the critical line.

Computational Implementation

The following Wolfram Language code implements the integral model for f(s) and compares its magnitude to the completed zeta function, Xi(s), to visualize the zero-valleys on the critical line.

Wolfram Code for Zeta-Kernel Visualization:

• ClearAll["Global`*"];
• Xi[s_] := (1/2) s (s - 1) Pi^(-s/2) Gamma[s/2] Zeta[s];
• fModel[s_?NumericQ, tMax_: 30] := Module[{x, y, pref, int1, int2, integrand1, integrand2}, x = Re[s]; y = Im[s]; pref = 4 * ((1 - x - I y)/(1 + x + I y)); integrand1[t_?NumericQ] := Module[{den, u, phase}, den = Exp[2 Pi t] - 1; u = Sqrt[1 + t^2]; phase = u^(-(x + I y)); (Sin[x ArcTan[t]] Cosh[y ArcTan[t]]/den) * phase]; integrand2[t_?NumericQ] := Module[{den, u, phase}, den = Exp[2 Pi t] - 1; u = Sqrt[1 + t^2]; phase = u^(-(x + I y)); (Cos[x ArcTan[t]] Sinh[y ArcTan[t]]/den) * phase]; int1 = NIntegrate[integrand1[t], {t, 0, tMax}, Method -> "GlobalAdaptive", WorkingPrecision -> 20]; int2 = NIntegrate[integrand2[t], {t, 0, tMax}, Method -> "GlobalAdaptive", WorkingPrecision -> 20]; pref*int1 + 4 I * ((1 - x - I y)/(1 + x + I y)) * int2];
• Plot[{Abs[fModel[1/2 + I y, 20]], Abs[Xi[1/2 + I y]]}, {y, 10, 30}, PlotLegends -> {"|fModel|", "|Xi|"}, PlotStyle -> {Red, {Blue, Dashed}}]

Conclusions

The synthesis of discrete magma algebra and oscillatory integral kernels provides a robust diagnostic tool for the Riemann Hypothesis. By analyzing the positivity of Hankel matrices derived from the moment expansions of f(s), researchers can pinpoint the exact conditions required for the zeros to remain on the critical line. Future research should focus on proving the non-negativity of the kernels g0 and g1 under appropriate normalizations, which would formally establish the necessary constraints on the zeta function's zeros.