Extension to Alternating Sums: Rapidly Convergent Summation Formulas for Dirichlet Eta, Beta, and General Alternating L-Functions at Odd Positive Integers

This paper systematically extends the recently established rapidly convergent summation formula for the odd-order Riemann zeta function ζ(2m + 1), along with its proof techniques, to alternating sum forms of Riemann-type functions. The primary focus is on the Dirichlet eta function η(s), the Dirichlet beta function β(s), and general periodic alternating L-functions L(s,χ). We provide explicit summation formulas valid at odd positive integer arguments for these functions. Each formula consists of a rational-coefficient term involving a power of π plus a rapidly convergent infinite series involving Bernoulli numbers (or generalized Bernoulli numbers). Rigorous proofs are supplied using a multi-tiered approach: an elementary proof requiring only calculus and an advanced proof employing generating functions and complex integration techniques. High-precision numerical computations are performed to verify the correctness and efficiency of the formulas. Furthermore, we establish a unified integral representation connecting these alternating sum formulas with the classical Euler formula for even zeta values, revealing an underlying symmetry at the generating function level. A profound analogy with Fermi-Dirac statistics in quantum statistical physics is also explored. The results provide new analytical tools and a rigorous mathematical framework for the theoretical analysis and high-precision computation of alternating Riemann-type functions.