An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
@article{Vepstas2007AnEA, title={An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions}, author={Linas Vepstas}, journal={Numerical Algorithms}, year={2007}, volume={47}, pages={211-252} }
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function” by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating Lis(z) for general…
37 Citations
Fast and Rigorous Computation of Special Functions to High Precision
- Computer Science, Mathematics
- 2014
This work gives new baby-step, giant-step algorithms for evaluation of linearly recurrent sequences involving an expensive parameter and for computing compositional inverses of power series and shows that isolated values of the integer partition function p(n) can be computed rigorously with softly optimal complexity by means of the Hardy-RamanujanRademacher formula and careful numerical evaluation.
The Polylogarithm Function in Julia
- Computer ScienceArXiv
- 2020
This paper presents an algorithm for calculating polylogarithms for both complex parameter and argument and evaluates it thoroughly in comparison to the arbitrary precision implementation in Mathematica.
Rigorous high-precision computation of the Hurwitz zeta function and its derivatives
- MathematicsNumerical Algorithms
- 2014
New record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function are presented using an open source implementation of the algorithms described in this paper.
High-precision computation of uniform asymptotic expansions for special functions
- Mathematics, Computer Science
- 2019
New methods to obtain uniform asymptotic expansions for the numerical evaluation of special functions to high-precision are investigated, obtaining efficient new convergent and uniform expansions for numerically evaluating the confluent hypergeometric functions and the Lerch transcendent at high- Precision.
Series with Binomial-Like Coefficients for Evaluation and 3D Visualization of Zeta Functions
- MathematicsInformatica
- 2020
A central limit theorem is proved for the coefficients of the series with binomial-like coefficients used for evaluation of the Riemann zeta function and the rate of convergence to the limiting distribution is established.
Series of Floor and Ceiling Functions—Part II: Infinite Series
- MathematicsMathematics
- 2022
In this part of a series of two papers, we extend the theorems discussed in Part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta…
The Riemann Hypothesis: A Qualitative Characterization of the Nontrivial Zeros of the Riemann Zeta Function Using Polylogarithms
- Mathematics
- 2016
We formulate a parametrized uniformly absolutely globally convergent series of ζ(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s − 1)ζ(s) + 1 x Li s z z − 1 dz,…
Modern Computer Arithmetic
- Computer Science
- 2010
Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics…
THE BERNOULLI OPERATOR
- Mathematics
- 2014
This document explores the Bernoulli operator, giving it a variety of different definitions. In one definition, it is the shift operator acting on infinite strings of binary digits. In another…
A new non-negative distribution with both finite and infinite support
- Mathematics
- 2020
The Tukey-$\lambda$ distribution has interesting properties including (i) for some parameters values it has finite support, and for others infinite support, and (ii) it can mimic several other…
References
SHOWING 1-10 OF 47 REFERENCES
Resurgence of the fractional polylogarithms
- Mathematics
- 2007
The fractional polylogarithms, depending on a complex parameter �, are defined by a series which is analytic inside the unit disk. After an elementary conversion of the series into an integral…
Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series
- Computer Science
- 1989
Polynomial Expansions of Analytic Functions
- Mathematics
- 1958
This is an excellent little book written with a commendable attention to detail. The authors take pains to indicate frequently what is going on behind the scenes and they include numerous pertinent…
An efficient algorithm for the Riemann zeta function
- Computer Science
- 1995
A very simple class of algorithms for the computation of the Riemann-zeta function to arbitrary precision in arbitrary domains is proposed. These algorithms out perform the standard methods based on…
Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions
- Mathematics
- 2007
Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by…
Introduction to analytic number theory
- Mathematics
- 1976
This is the first volume of a two-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years. It provides an introduction…
Structural Properties of Polylogarithms
- Mathematics
- 1991
The evolution of the ladder concept by L. Lewin Dilogarithmic ladders by L. Lewin Polylogarithmic ladders by M. Abouzahra and L. Lewin Ladders in the trans-Kummer region by M. Abouzahra and L. Lewin…