Hypergeometric Deformations and the Adelic Geometry of Zeta Zeros

Introduction

The search for a proof of the Riemann Hypothesis has historically oscillated between purely analytic number theory and spectral geometry. The source paper arXiv:hal-01114315 provides a bridge between these domains by constructing a global analytic geometry based on weak Mellin transforms of second-degree characters. This framework recontextualizes the Riemann zeta function as a global object emerging from local distributions over archimedean and non-archimedean fields.

The central motivation of this analysis is the study of quadratic characters, specifically functions of the form f(x) = exp(i pi (ax^2 + bx)). The paper demonstrates that the local zeta functions associated with these characters can be expressed through Kummer confluent hypergeometric functions. This is significant because the distribution of zeros for such functions offers a controlled environment to study the stability of the critical line under geometric transformations of parameters a and b.

By unifying p-adic and real components through an operator-theoretic lens, the research identifies how the Pochhammer identities and adelic integrals found in arXiv:hal-01114315 offer a novel mechanism for localizing the zeros of the global zeta function. This article explores the spectral properties of these underlying spaces and proposes new pathways for establishing the location of non-trivial zeros.

Mathematical Background

The primary mathematical object in the analysis is the weak Mellin transform, denoted as zeta_a,b(s). Defined for a quadratic character over a field K, this transform leads to integral representations that, in the real case, are defined as distributions. A key identity established in the paper links spatial deformation (parameter b) to scale deformation (parameter a):

The k-th derivative with respect to the shift parameter b, evaluated at b=0, satisfies the relation: (d/db)^2k zeta_a,b(s) = (-4 pi i)^k (d/da)^k zeta_a,b(s). This identity allows the zeta function to be expressed using the rising factorial or Pochhammer symbol, (s/2)_k.

In the non-archimedean (p-adic) setting, the paper examines integrals over Z_p (p-adic integers) and Q_p (p-adic numbers). The relationship zeta_f(s, chi) = (Nd)^(-1/2) (1 - q^-s)^-1 establishes the local factor of the zeta function, where q is the cardinality of the residue field. The interplay between these local factors across all primes forms the basis of a global analytic geometry where the Riemann Hypothesis can be framed as a question of local-global compatibility.

Main Technical Analysis

Hypergeometric Representation and Parameter Deformation

The core technical innovation in arXiv:hal-01114315 is the representation of the local zeta function as a Kummer confluent hypergeometric function 1F1. For the real case, the transform is proportional to Gamma(s/2) 1F1(s/2, 1/2, i pi b^2 / 2a). This identifies a natural family of Mellin transforms interpolating between pure Gaussians (b = 0) and oscillatory phases (b != 0).

The Taylor expansion of this function around b=0 reveals a series involving Pochhammer symbols and powers of (2 pi i / a). The convergence of this series in the distributional sense allows the zeta function to be extended into regions where classical Dirichlet series diverge. In this context, the zeros of the transform correspond to the eigenvalues of a Hermite-type operator that vanish under the Mellin projection.

P-adic Integration and Euler Factor Emergence

The analysis extends to p-adic integrals that compare oscillatory behavior across different shells of the p-adic domain. By subtracting the integral over the maximal ideal pZ_p from the integral over the whole ring Z_p, the paper isolates a primitive contribution proportional to (1 - p^(s-n)). This demonstrates that the Euler factors of the Riemann zeta function emerge naturally from the geometry of p-adic space.

This result implies that the zeros of the global zeta function are the points where the product of these local p-adic distributions and the archimedean confluent hypergeometric function cancel out. This provides a geometric explanation for the multiplicative structure of the completed zeta function xi(s).

Functional Symmetry and Gamma Stability

The paper utilizes Gamma function identities to prove the functional equation of the local zeta function. Since the Fourier transform of a Gaussian is another Gaussian, the Mellin transform (the Gamma function) must satisfy a reflection formula. The paper proves that this symmetry s to 1-s is stable even when the character is deformed by parameters a and b, providing a robust framework for investigating the critical line Re(s) = 1/2.

Novel Research Pathways

Pathway 1: Hypergeometric Zero Trajectories

The representation of the zeta factor as 1F1(s/2, 1/2, z) allows for the study of zero dynamics. By investigating the trajectories of the zeros of this function as the complex parameter z moves, researchers can utilize homotopy methods to trace these points. If the zeros are constrained to the critical line for a specific domain of z, the limit as z approaches zero could offer a proof path for the Riemann Hypothesis.

Pathway 2: B-Deformed Explicit Formulas

The derivative identities in b provide a family of test functions for Weil's explicit formula. By summing the paper's Taylor series into explicit hypergeometric forms, one can analyze the sign and decay of these transforms on the critical line. This enables a systematic search for positivity criteria, where the Riemann Hypothesis is equivalent to the positive definiteness of certain archimedean weights.

Pathway 3: Adelic Trace Formulas

The action of GL_n(Z_p) on the space of functions suggests constructing a global Trace Formula using the weak Mellin transform as the fundamental kernel. The zeros of the zeta function would then relate to the eigenvalues of the generator of the deformation in the space of weak transforms, providing the spectral interpretation sought by the Polya-Hilbert conjecture.

Computational Implementation

The following Wolfram Language code demonstrates the relationship between the confluent hypergeometric function 1F1 and the zeros of the Riemann zeta function, modeling the archimedean local factor described in arXiv:hal-01114315.

(* Section: Hypergeometric Zeta Deformation *)
(* Purpose: Visualize the 1F1 proxy for the local zeta factor *)

ClearAll[localZeta, a, b];
a = 1.0; 
b = 0.15; 

(* Define local zeta factor using Equation 3.158 from the paper *)
localZeta[s_] := Exp[-s * Pi * I / 4] * (a^(-s/2)) * 
  (Pi^(-s/2)) * Gamma[s/2] * 
  Hypergeometric1F1[s/2, 1/2, I * Pi * b^2 / (2*a)];

(* Compare magnitude on the critical line Re(s)=1/2 *)
Plot[Abs[localZeta[1/2 + I*t]], {t, 0, 40}, 
  PlotRange -> All, 
  PlotStyle -> Thick, 
  Frame -> True, 
  FrameLabel -> {"Im(s)", "|zeta_a,b(s)|"}, 
  PlotLabel -> "Local Weak Mellin Transform Magnitude"];

(* Find pseudo-zeros of the deformed factor *)
pseudoZeros = Table[
  t /. FindRoot[Re[localZeta[1/2 + I*t]] == 0, {t, Im[ZetaZero[k]]}], 
  {k, 1, 5}
];

Print["Deformed Pseudo-Zeros (t-values): ", pseudoZeros];
Print["Actual Riemann Zeros: ", Table[Im[ZetaZero[k]], {k, 1, 5}]];

(* Complex phase visualization *)
ComplexPlot[localZeta[s], {s, 0 + 0I, 1 + 40I}, 
  ColorFunction -> "Phase", 
  PlotLabel -> "Phase Map of Deformed Local Factor"]

Summary of key findings: The deformation parameter b shifts the zeros of the local factor. As b approaches zero, the function reduces to the standard archimedean factor, which provides the necessary poles for the global zeta function to resolve.

Conclusions

The framework provided by Frédéric Paugam reveals that the Riemann Hypothesis is deeply connected to the symmetry of Fourier transforms over adeles. By defining the zeta function as a weak Mellin transform of quadratic characters, the critical line emerges as a geometric necessity. The most promising avenue for further research is the study of hypergeometric deformations, where the zeros of the zeta function are understood as fixed points of a dynamical system acting on the parameters of the underlying characters. Future steps should involve computing the spectral gap for the p-adic operators and extending the computational models to include multi-prime sieve products.

References

• arXiv:hal-01114315: Frédéric Paugam, "Global Analytic Geometry and the Riemann Hypothesis."
• Tate, J. T. (1950). "Fourier analysis in number fields, and Hecke's zeta-functions." PhD Thesis.
• Iwaniec, H. & Kowalski, E. (2004). "Analytic Number Theory." American Mathematical Society.