Mellin Kernel Fixed Point Methods for the Computation of Non-Trivial Zeros of the Riemann Zeta Function

Abstract

The non-trivial zeros of the Riemann zeta function remain fundamental objects in analytic number theory. In this paper, an enhanced Mellin transform framework is developed in which hyperbolic kernels generate spectral operators naturally associated with critical-line evaluations of the zeta function. Particular emphasis is placed on the kernels sech(x) and 1/ sinh x, whose Mellin transforms respectively provide numerical regularity and explicit arithmetic connections with ζ(s). A fixed-point operator is constructed for the ordinate parameter t corresponding to points s = 21 + it on the critical line. Using complete metric intervals, compactness of closed bounded sets, derivative-based Lipschitz estimates, and the Banach contraction principle, local and global convergence criteria are established. It is further shown that every simple critical-line zero induces a locally unique contractive fixed point. A direct bridge with the Hardy Z-function and the Riemann–Siegel framework is obtained through Mellin spectral coordinates. Numerical experiments demonstrate stable convergence for benchmark ordinates ranging from low-lying zeros to substantially higher-order zeros. The framework also suggests an operator-theoretic pathway for investigating the Rie-mann Hypothesis through completeness of Mellin residual roots on the critical line. The resulting formulation combines Mellin transforms, hyperbolic kernels, and nonlinear dynamics into a compact method for critical-line zero approximation.