Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complex Moments, Gamma and Riemann Zeta Functions unified by the Parabolic Mellin Transform

Peter Reinhard Hansen, Chen Tong

Abstract

We present a unified integral framework based on the Fourier-Laplace transform evaluated along a vertical line in the complex plane. By identifying the moment-generating function (MGF) of a random variable with the weights of these integrals, we first establish a general expression for complex fractional moments valid for any random variable with a MGF. Applying this formula to the Gaussian distribution, we recover a global integral representation for the reciprocal Gamma function that unifies it with its reflection. We formalize the underlying operator as the Parabolic Mellin Transform, a holomorphic alternative to the classical Mellin transform that avoids strips of convergence by mapping the vertical line to a parabolic contour. This general framework leads to new meromorphic representations for the Hurwitz and Riemann zeta functions that are valid throughout the critical strip, as well as reformulations of the Riemann hypothesis and the Lindelof hypothesis.

Complex Moments, Gamma and Riemann Zeta Functions unified by the Parabolic Mellin Transform

Abstract

We present a unified integral framework based on the Fourier-Laplace transform evaluated along a vertical line in the complex plane. By identifying the moment-generating function (MGF) of a random variable with the weights of these integrals, we first establish a general expression for complex fractional moments valid for any random variable with a MGF. Applying this formula to the Gaussian distribution, we recover a global integral representation for the reciprocal Gamma function that unifies it with its reflection. We formalize the underlying operator as the Parabolic Mellin Transform, a holomorphic alternative to the classical Mellin transform that avoids strips of convergence by mapping the vertical line to a parabolic contour. This general framework leads to new meromorphic representations for the Hurwitz and Riemann zeta functions that are valid throughout the critical strip, as well as reformulations of the Riemann hypothesis and the Lindelof hypothesis.
Paper Structure (19 sections, 13 theorems, 82 equations, 3 figures, 1 table)

This paper contains 19 sections, 13 theorems, 82 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. General Theory
  3. Integral Representation of Moments
  4. Connection to Special Functions
  5. The Gamma Function and its Reciprocal
  6. Riemann and Hurwitz zeta Functions
  7. Parabolic Geometry and the Hankel Contour
  8. Symmetrized Integrals and the Critical Line
  9. The Parabolic Mellin Transform
  10. Definition and Geometry
  11. Properties of Parabolic Mellin Transform
  12. Table of Transforms
  13. Conclusion
  14. Proofs of Auxiliary Results
  15. Vanishing Identity
  16. ...and 4 more sections

Key Result

Lemma 1

Let $w = \sigma + it$ with $\sigma > 0$. For any real $x \neq 0$ and any $r \in \mathbb{C}$ with $\operatorname{Re}(r) > -1$, we have the identity:

Figures (3)

  • Figure 1: The image of the vertical line $w=\sigma+it$ under the mapping $w\mapsto u=w^2$. The resulting parabolic contour $t\mapsto (\sigma^2-t^2,2i\sigma t)$ wraps around the origin at a distance $\sigma^2$, avoiding the branch cut on the negative real axis. As $t \to \pm \infty$, the real part of $u=w^2$ tends to $-\infty$, enforcing rapid decay of $e^{\alpha w^2}$ for $\alpha>0$.
  • Figure 2: The integration contour $C_R$ in the complex $t$-plane. The path along the real axis $[-R, R]$ is closed by a semi-circle in the lower half-plane. The singularity at $t=i\sigma$ lies outside the contour.
  • Figure 3: The closed contour $\mathcal{C}$ used in the Vanishing Lemma. The path $A \to B$ represents the integration along $w = \sigma + it$. The contour is closed via arcs to the imaginary axis ($B \to C$ and $F \to A$) and a small indentation $D \to E$ to avoid the singularity at the origin. The teal circles indicate the poles of the integrand.

Theorems & Definitions (29)

  • Lemma 1: Integral Representation of Absolute Powers
  • proof
  • Remark 1
  • Theorem 1: Absolute Moments
  • proof
  • Theorem 2: Unified Gamma Representation
  • proof
  • Remark 2
  • Corollary 1: Integral Representations for $\psi(z)$ and $\gamma$
  • Theorem 3: Integral Representation of zeta Functions
  • ...and 19 more