The Parabolic Mellin Transform: Gamma and Zeta Integral Representations

Technical Summary: The Parabolic Mellin Transform: Gamma and Zeta Integral Representations

Problem Statement
The paper addresses the representation of special functions, specifically the Gamma function, its reciprocal, and various zeta-type functions (Hurwitz, Riemann, and Dirichlet eta), through contour integration. While classical representations often rely on the Hankel contour (a keyhole contour wrapping around the negative real axis) or the Laplace integral (a vertical line integral), these methods have distinct limitations. The Hankel contour is geometrically complex, and the Laplace integral for the reciprocal Gamma function relies on oscillatory cancellation, leading to conditional convergence in parts of its domain. The authors seek a unified framework that expresses these functions via a vertical-line integral with absolute convergence, leveraging Gaussian decay to simplify analysis and provide a common notation for Gamma and zeta functions.

Methodology
The core methodology introduces the Parabolic Mellin Transform (PMT), defined by the quadratic change of variables $u = w^2$.

1. Definition: For a function $f$ defined on the image of the map $w \mapsto w^2$, the PMT is defined as:
   $$P_\sigma[f](z) = \int_{-\infty}^{\infty} w^{2z} f(w^2) \, dt, \quad \text{where } w = \sigma + it, \sigma > 0.$$
   Under the substitution $u = w^2$, the vertical line $\text{Re}(w) = \sigma$ maps to a left-opening parabolic contour $C_\sigma$ in the $u$-plane with vertex at $\sigma^2$.
2. Gaussian Damping: A critical feature of this parametrization is that exponential kernels $e^u$ become $e^{w^2} = e^{\sigma^2 - t^2}$. This transforms the contour integral into a vertical-line integral with Gaussian damping ($e^{-t^2}$), ensuring absolute convergence for all $z \in \mathbb{C}$ and allowing for holomorphy under the integral sign.
3. Contour Deformation: The authors utilize Cauchy's theorem to deform the parabolic contour $C_\sigma$ to the standard Hankel contour. This establishes the equivalence between the PMT and classical Hankel representations without altering the analytic continuation properties of the functions involved.
4. Scaling and Composition: The framework employs scaling identities ($\alpha^{-s}$) and Dirichlet composition to relate the Gaussian transform to Dirichlet series, thereby connecting the transform to zeta functions.

Key Contributions and Results

• The Gaussian Transform ($G(z)$):
  The paper proves that the PMT of the Gaussian kernel $e^u$ yields an entire function $G(z)$ that represents both the Gamma function and its reciprocal at reflected arguments. Specifically, with $s = z + 1/2$:
  $$G(z) = \frac{\pi}{\Gamma(1-s)} = \Gamma(s)\sin(\pi s).$$
  This provides a parabolic-contour form of the classical Hankel representation for $1/\Gamma(s)$. Unlike the Laplace integral, this representation is absolutely convergent for all $z$.

• Zeta-Type Integral Representations:
  By combining the Gaussian transform with meromorphic kernels such as $\frac{e^{au}}{1-e^u}$ and $\frac{e^{au}}{1+e^u}$, the authors derive integral representations for the Hurwitz zeta function $\zeta(s, a)$, the Riemann zeta function $\zeta(s)$, and the Dirichlet eta function $\eta(s)$.
  The fundamental identities are:
  $$R(z, a) = G(z)\zeta(s, a) \quad \text{and} \quad D(z, a) = G(z)\eta(s, a),$$
  where $R$ and $D$ are the PMTs of the respective kernels. These identities hold meromorphically, with the zeros of $G(z)$ (at $z = 1/2 + n$) corresponding to the poles of the zeta functions or removable singularities.

• A Unified Transform Dictionary:
  The paper constructs a "dictionary" (Table 1) that unifies Gamma-type and zeta-type functions under a single transform notation. This dictionary includes operations for scaling, monomial shifts, differentiation, and logarithmic shifts, all expressed within the PMT framework.

• Reformulation of Critical Strip Problems:
  The paper demonstrates how the PMT coordinates ($z = s - 1/2$) allow for exact reformulations of major conjectures:

  • Riemann Hypothesis (RH): Since $G(z)$ has no zeros in the strip $|\text{Re}(z)| < 1/2$, the non-trivial zeros of $\zeta(s)$ correspond exactly to the zeros of $R(z)$ in this strip. Thus, the RH is equivalent to the assertion that all zeros of $R(z)$ in $|\text{Re}(z)| < 1/2$ lie on the imaginary axis ($\text{Re}(z) = 0$).
  • Lindelöf Hypothesis: The hypothesis is reformulated as a growth condition on the parabolic integral $R(i\tau)$. Specifically, since $|G(i\tau)| \sim \sqrt{\pi} e^{\pi|\tau|/2}$, the Lindelöf hypothesis is equivalent to $R(i\tau) = O_\epsilon(e^{\pi|\tau|/2}(1+|\tau|)^\epsilon)$.

Significance and Claims
The authors explicitly state that the PMT does not provide a new analytic continuation of the Gamma or zeta functions, nor does it yield new zero-free regions or improved growth bounds. Instead, its significance lies in reorganization and unification:

1. Parametrization: It expresses classical Hankel-type integrals through a parabolic parametrization ($u=w^2$) that converts them into Gaussian-damped vertical-line integrals.
2. Convergence: It replaces conditionally convergent oscillatory integrals (like the Laplace formula) with absolutely convergent Gaussian-damped integrals.
3. Unification: It provides a common normalizing factor ($G(z)$) for both Gamma-type and zeta-type identities, centering the critical strip at zero and simplifying the notation for comparing these functions.
4. Equivalence: It offers equivalent reformulations of the Riemann and Lindelöf hypotheses in terms of the zeros and growth of parabolic-contour integrals, isolating the explicit exponential factors to focus on the remaining integral behavior.

The paper concludes that the PMT serves as a "transform dictionary" that facilitates the comparison of Gamma-type normalizations, zeta-type kernels, and critical-line growth within a single analytic notation.