Fast Convergent Summation Formulas for the Riemann Zeta Function of Real Orders and Their Generalizations

This paper extends the known summation formulas for the Riemann zeta function at odd integer orders to real orders (s > 1). By replacing factorials with Gamma functions, we establish fast convergent summation formulas suitable for real orders. We first present an integral representation for the zeta function of real order and rigorously prove its equivalence to a series representation. The formulas unify the cases of odd and even integer orders,revealing the intrinsic relationships among the zeta function, the constant π, and Bernoulli numbers. We provide complete mathematical derivations, including convergence analysis and special value verification. Numerical experiments demonstrate that the formulas exhibit super-exponential convergence; for s > 2, typically only the first 10–20 terms are needed to achieve double-precision accuracy. Furthermore, we prove that the formulas also hold for 0 <s≤1, thereby offering a new perspective on analytic continuation and discussing their potential value in computation and applications.