Abstract
Polynomial expansions of zeta functions provide a natural way to connect analytic continuation, regularized summation, Mellin analysis and orthogonal polynomial theory. In this paper we try to develop a shifted Ramanujan–Mellin expansion for the Hurwitz zeta function in the critical strip. The construction combines Abel–Plana regularization over the nonnegative integers, Ramanujan summation for shifted Dirichlet terms, the Cayley transform of the right half-plane, and Mellin transforms of Laguerre functions. The main result proves that the Hurwitz zeta function admits a locally uniformly convergent expansion in a universal polynomial basis that is independent of the shift parameter. The shift appears only through explicit coefficients involving the digamma function and shifted Hurwitz zeta values. The Riemann zeta function is obtained as a special case. On the critical line, the normalized basis forms a complete orthonormal system with respect to a hyperbolic weight, and every zero of each basis polynomial lies on the critical line. The final result gives a precise zero-free compact criterion equivalent to the Riemann Hypothesis.
