An efficient algorithm for computing Dirichlet<i>L</i>-functions

Šleževičienė, R.

doi:10.1080/1065246042000272072

Cited by 5 publications

(5 citation statements)

References 3 publications

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“…For the Borwein algorithm, evaluating the bounds (24) and (25), so as to obtain an order estimate, is non-trivial, and can account for a significant amount of compute time. A speedier heuristic is desirable.…”

Section: Resultsmentioning

confidence: 99%

“…To obtain a value of the polylogarithm to within some given numerical precision, one must invert the formulas (24) or (25), solving for the value of n which is to be used in (23). To obtain a fixed number of digits of precision, one must carry out intermediate calculations with at least that many digits of precision.…”

Section: Bound On the Error Termmentioning

confidence: 99%

“…There has been little work in this area. Šleževičiene provides a discussion of fast algorithms for Dirichlet L-functions in [24]. Balanzario presents an extension of the Euler-Maclaurin formula, such that it can be used to evaluate ζ(s, q) in the critical strip [3].…”

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confidence: 99%

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An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

Vepstas

2008

Numer Algor

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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 Li s (z) for general values of complex s and a kidney-shaped region of complex z values given by z 2 /(z − 1) < 4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included. L. Vepštas (B)

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Section: Resultsmentioning

confidence: 99%

Section: Bound On the Error Termmentioning

confidence: 99%

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An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

Vepstas

2008

Numer Algor

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“…In view of (14), to compute the Riemann zeta-function with d decimal digits of accuracy, the approach requires a number n of terms not less than…”

Section: γmentioning

confidence: 99%

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

Belovas¹

2022

Preprint

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The paper continues the study of efficient algorithms for the computation of zeta functions over the complex plane. We aim to apply the modifications of algorithms to the investigation of underlying fractal structures associated with the Riemann zeta function. We discuss the computational complexity and numerical aspects of the implemented algorithms based on series with binomial-like coefficients.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2022

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An efficient algorithm for computing DirichletL-functions

Cited by 5 publications

References 3 publications

An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

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

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

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