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An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
Classical Analysis and ODEs · Submitted 9 Feb 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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", to more general series. The algorithm provides a rapid means of evaluating Li_s(z) for general values of complex s and the region of complex z values given by |z^2/(z-1)|<4. 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.98 | 0.89 | 0.15 | asserted 0.98 |
| s1 | 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", to more general series. | background 0.80 | 0.17 | 0.05 | hedged possibility 1.00 |
| s2 | The algorithm provides a rapid means of evaluating Li_s(z) for general values of complex s and the region of complex z values given by |z^2/(z-1)|<4. | contribution 0.61 | 0.77 | 0.11 | asserted 1.00 |
| s3 | Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler-Maclaurin series. | background 0.75 | 0.12 | 0.04 | asserted 0.71 |
| s4 | 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. | background 0.49 | 0.31 | 0.06 | asserted 0.99 |
| s5 | 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. | result 0.85 | 0.76 | 0.14 | asserted 1.00 |
| s6 | Both algorithms are superior to the simple Taylor's series or direct summation. | result 0.87 | 0.66 | 0.06 | asserted 1.00 |
| s7 | 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. | contribution 0.53 | 0.84 | 0.92 | asserted 1.00 |
| s8 | A discussion of the monodromy group of the polylogarithm is included. | contribution 0.98 | 0.65 | 0.02 | asserted 0.99 |
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Original abstract
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", to more general series. The algorithm provides a rapid means of evaluating Li_s(z) for general values of complex s and the region of complex z values given by |z^2/(z-1)|<4. 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.
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4 arXiv papers cite this in 6 sentences
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Uses its method or tool 2 papers
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The continuous branch extension of $(1,)$ is given by [ref] (1,)=(1)+ 8/3 π^4 ∑_i (n_i)^3 = π^4/90 +8/3 π^4 ∑_i (n_i)^3 , n_i ∈ , and the resulting modification to the Nambu-Goto spectrum turns out to be E_{n_i }^()(R,δ_2) = σR (2 κ-2)=E_n^()(R) + δ_2 π^3/4 R^4 ( 1/60 + 4 ∑_i (n_i)^3 ) +…
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According to Eq. [ref], the ground-state term can be estimated by $ζ(1/2,γ^2)$ in small- or large-$γ$ expansions [ref] .
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