A method for accelerated computation of the Riemann zeta function on the complex plane

Central Limit Theorems for Combinatorial Numbers Associated with Laguerre Polynomials

“…Apart from the theoretical value (generating functions are a very important tool to derive the identities, connections, and interpolation functions for polynomials, or limit theorems for corresponding combinatorial numbers), these results can be applied to the construction of efficient algorithms for the calculation of the values of special functions. We have used similar limit theorems for the combinatorial numbers in calculations of the Riemann zeta function (see Theorem 3 in [14] and Algorithm 3 in [17]). Moreover, the presented asymptotic normality results may have also an important utilization in choosing a suitable cumulative distribution function or a cumulative intensity function for models in insurance [18].…”

Series with Binomial-like Coefficients for the Investigation of Fractal Structures Associated with the Riemann Zeta Function

“…In this paper, we continue the study of efficient algorithms for computation of the Riemann zeta function over the complex plane, introduced by Borwein [1] and extended by Belovas et al, (see [2,3] and references therein). Šleževičien ė [4], Vepštas [5], and Coffey [6] applied this methodology for the computation of Dirichlet L-functions, Hurwitz zeta function, and polylogarithm.…”

“…Belovas et al obtained limit theorems, which allowed the introduction of asymptotic approximations for the coefficients of the series of the algorithms. A preliminary presentation of computational aspects of the approach has been presented in [3]. Theoretical aspects of the approach (as well as more subtle proofs of the limit theorems) have been discussed in [7].…”

Series with Binomial-Like Coefficients for the Investigation of Fractal Structures Associated with the Riemann Zeta Function

“…where s k p stand for critical points (5) with k p = 2 p+6 , 1 p 3, and ρ 1 = 10 −1 . Thus we obtain 9 sequences overall (3 algorithms × 3 sets of arguments).…”

“…Γ(s), B(x, y) and W(x) denote the gamma function, the beta function and the Lambert W function respectively. I x (a, b) stands for the regularized incomplete beta function, In [5] Belovas et al proposed a modification of Borwein's efficient algorithm (MBalgorithm) for the Riemann zeta function [7]. The algorithm applies to complex numbers s = σ + it with σ 1/2 and arbitrary t. Let us denote, along with Proposition 1 from [5],…”