Explicit Integral Equations for Zeros of Generalized L-Functions:A Complete Theory

Based on the contour integral representation method,we establish a complete theory of explicit integral equations for zeros of generalized L-functions satisfying functional equations and Euler products.We first present the most general contour integral representation theorem for L-functions,proving the convergence and analytic continuation of this representation.On this basis,we derive the integral equations satisfied by zeros and the irequivalent forms, establishing the characterization of zero distribution via integral operators.Further more,we extend this framework to a wide range of cases including Selberg class functions,automorphic representation L-functions,arithmetic zeta functions, and operator zeta functions,constructing a unified theoretical system. This theory provides universal analytic tools for studying zero distributions of various zeta functions,and presents corresponding numerical algorithm frameworks with error analysis.The integral equation approach offers advantages over classical explicit formulas in terms of numerical stability,avoidance of conditional convergence,and facilitating operator-theoretic treatments of zero distributions.As a significant application,we derive a fully-integral version of the explicit formula for primenumber distribution by combining the zero integral equations with the Perron formula,transforming the conditional sum over zeros into a finite-path integral with enhanced numerical stability.