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.