Closed-Form Expressions, Best Approximation Theory, and Arithmetic Properties of Dirichlet L-Functions of Real Order

Based on the theory of closed-form expressions for the Riemann ζ function of real order, this work systematically extends it to general Dirichlet L-functions. By introducing tools such as the analytic continuation of the Γ function, fractional calculus, and generalized hypergeometric functions, a unified representation framework for Dirichlet L-functions of real order is established. Firstly, integral closed-form expressions for these functions are given, along with a convergence analysis. Secondly,a representation system based on generalized hypergeometric functions is constructed, providing explicit expressions for correction terms. Thirdly, the theory of Γ-rational approximation is proposed,refining error estimation methods. Fourthly, explicit closed-form expressions and recursive construction methods are provided for the half-integer order case. Fifthly, fractional differential equations satisfied by Dirichlet L-functions of real order are established. Additionally, this paper systematically studies the best approximation theory, irrationality, and algebraic independence of these functions. All theoretical results are accompanied by rigorous mathematical proofs and numerical verification.