The Parabolic Mellin Transform: Gamma and Zeta Integral Representations

We introduce the Parabolic Mellin Transform (PMT), defined by ${P}_{\sigma}[f](z)=\int_{-\infty}^{\infty}w^{2z}f(w^2)dt$, where $w=\sigma+it$ and $\sigma>0$. Under the substitution $u=w^2$, the vertical line $\operatorname{Re}(w)=\sigma$ is mapped to the parabolic contour $C_\sigma$ in the $u$-plane. For the Gaussian kernel, the PMT yields $\int_{-\infty}^{\infty}w^{2z}e^{w^2}dt=\pi/\Gamma(\tfrac{1}{2}-z)=\cos(\pi z)\Gamma(z+\tfrac{1}{2})$, a parabolic-contour form of the classical Hankel representation for the reciprocal Gamma function. The advantage of this parametrization is that the contour integral becomes a Gaussian-damped vertical-line integral. We develop scaling, differentiation, and Dirichlet-composition identities for the PMT and use them to derive integral representations of the Hurwitz zeta, Riemann zeta, and Dirichlet eta functions. The framework provides a unified transform dictionary for Gamma-type and zeta-type special functions and yields equivalent reformulations of the Riemann hypothesis and the Lindel\"of hypothesis in terms of zeros and growth of parabolic-contour integrals.