A Geometric Framework for Zero Counting in Dirichlet L-Function Families

This paper investigates the analytic structure of the family of Dirichlet Lfunctions on the critical line from a geometric perspective, proposing a zero-counting mechanism based on geometric structure. By introducing a unitary normalization factor W (χ) -1/2 , we construct a real-valued function Ξ Λ (s, χ) with conjugate symmetry. On this basis, we establish the zero-counting principle of "Anchoring at Two Ends, Counting in the Middle," and derive an integer-type counting formula independent of the asymptotic error term S(T, χ): In addition, this paper applies the proposed universal geometric counting framework to an extended context, demonstrating that it naturally encompasses the classical case of the Riemann ζ(s) function.