What question did this study set out to answer?

The aim is to extend summation formulas for the Riemann zeta function to complex orders, proving convergence and revealing connections with other mathematical concepts.

February 2, 2026Open Access

Rapidly Convergent Summation Formulas for the Riemann Zeta Function of Complex Order and Multivariate Generalizations with Extensions

Key Points

The aim is to extend summation formulas for the Riemann zeta function to complex orders, proving convergence and revealing connections with other mathematical concepts.
Establish integral representation for complex order zeta function.
Generalize factorial to Gamma function for rapid convergence.
Conduct numerical experiments to validate accuracy and convergence rate.
Explore extensions to Hurwitz zeta function and multiple zeta values.
The series converges rapidly, needing only 10-20 terms for double-precision accuracy when Re(s) > 2.
The formula is valid for 0 < Re(s) ≤ 1, offering insights into analytic continuation.
Extensions to the entire complex plane include connections with modular forms and applications in physics.

Abstract

This paper extends the known summation formulas for the Riemann zeta function of odd positive integer order to complex order. By generalizing the factorial to the Gamma function, we establish rapidly convergent summation formulas valid for Re(s) > 0. We first present an integral representation for the zeta function of complex order and rigorously prove its equivalence to a series representation. The formula unifies the cases of both odd and even integer orders, revealing the intrinsic connections between the zeta function, the constant π, and Bernoulli numbers. We provide complete mathematical derivations, including convergence analysis and validation at special values. Numerical experiments demonstrate that the series converges with super-exponential rate; for Re(s) > 2, only the first 10–20 terms are typically needed to achieve double-precision accuracy. Furthermore, we prove that the formula remains valid in the region 0 < Re(s) ≤ 1, thereby offering a new perspective on analytic continuation. We systematically extend the formula to the whole complex plane (including Re(s) ≤ 0), the Hurwitz zeta function (with special attention to numerical stability for 0 < a <1), multiple zeta values (obtaining rapidly convergent representations via generating functions and integral transforms), the Epstein zeta function, q-analogues, and quantum zeta functions.Efficient numerical algorithms based on these formulas are developed and comprehensively compared with existing methods (such as the Riemann–Siegel formula and Euler–Maclaurin summation). We explore deep connections of the formulas with modular forms and demonstrate their effectiveness in analytic number theory (e.g., prime distribution, class number formulas, the Dirichlet divisor problem, and searches for Landau–Siegel zeros) and in physical applications (quantum field theory, statistical physics, string theory). This work not only provides new tools for computing special functions but also reveals intrinsic links among different mathematical fields.

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