This paper extends the known summation formulas for odd positive integer order Riemann zeta function to real orders s > 1 by generalizing the factorial to the Gamma function. We first present an integral representation for the alternating Riemann function of real order (Dirichlet eta function) and rigorously prove its equivalence to the series representation. The formula unifies the even and odd order cases,revealing the intrinsic connections among the eta function, the circle constant π, and Bernoulli numbers.Furthermore, we systematically generalize this method to the alternating Hurwitz zeta function (Hurwitz eta function) with parameters, obtaining corresponding rapidly convergent series representations.This paper provides complete mathematical derivations, including convergence analysis and special value verification. Numerical experiments demonstrate that the proposed formulas exhibit super-exponential convergence: for s > 2, typically only the first 10–20 terms are needed to achieve double precision; for 0 <s≤1,theformulas remain effective, thus offering new perspectives for analytic continuation. We also discuss in detail the potential value of these formulas in computation and applications, comparing them with existing algorithms such as the Riemann–Siegel formula. The main innovation of this paper lies in comprehensively extending the rapid convergence method from the standard zeta function to alternating Riemann-type functions, providing a unified framework for high-precision computation.
shifa liu (Wed,) studied this question.