This paper begins with the known rapidly convergent series for the Ap´ery constant ζ(3) and, through structural induction and generating function methods, proposes an explicit summation formula applicable to all odd-order Riemann zeta functions ζ(2m+1), presenting it as a theorem with rigorous mathematical formulation.We provide a complete proof based on generating function theory and complex analysis, establishing a rigorous mathematical foundation for this formula. Simultaneously, we systematically verify the formula for m = 1 to 7 (i.e., ζ(3) to ζ(15)) using high-precision numerical calculations, achieving errors on the order of 10−28 with the first 100 terms, confirming the formula’s correctness and rapid convergence. Furthermore, we explore the intrinsic unity between this odd-order formula and the even-order Euler formula, presenting a unified integral representation covering both parity cases, revealing the essential symmetry of odd and even zeta values at the generating function level. This paper also deeply explores elementary proofs, generalizations to Hurwitz zeta functions and multiple zeta values, connections with modular form L-functions, and potential applications in the BSD conjecture and Beilinson conjecture. The results provide new analytic tools and a rigorous math ematical framework for the theoretical analysis and high-precision computation of odd zeta values.
shifa liu (Wed,) studied this question.