Unified Theory of Closed-Form Expressions: From the Riemann Zeta Function to the L-Function Family

Based on the established theory of closed-form expressions for odd-order Riemann zeta functions and its “Rational Combination Conjecture” 1,2, this paper constructs a unified theoretical framework that systematically generalizes the theory to a broad family of functions, including the Dirichlet eta function, the Hurwitz eta function, the Lerch transcendent, and general Dirichlet L-functions. We first establish a parameterized fundamental representation theorem, then develop a unified family of integral closed-form expressions, sine integral expressions, and delve into the intrinsic connections between these functions and generalized Catalan constants. Ultimately, we prove and generalize the Unified Rational Combination Theorem: for a large class of L-functions at odd positive integer arguments, after appropriate normalization, they can all be expressed as rational functions of the circular constant π, and there exists an upper bound for the degrees of the numerator and denominator polynomials that depends only on the function’s parameters (for rational modulus parameters, the upper bound is 4q) and is independent of the order m. All results are accompanied by rigorous mathematical derivations and numerical verification.