Some rapidly converging series for $\zeta (2n+1)$

Srivástava, H. M.

doi:10.1090/s0002-9939-99-04945-x

Cited by 28 publications

(6 citation statements)

References 23 publications

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“…Here (and elsewhere in this work) an empty sum is to be interpreted (as usual) to be nil. We choose to recall the proof of (2.4) detailed by Srivastava [24]. Each of the other results (2.5), (2.6), and (2.7) can be proven mutatis mutandis.…”

Section: The First Set Of Series Representationsmentioning

confidence: 96%

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Some Families of Rapidly Convergent Series Representations for the Zeta Functions

Srivástava¹

2000

Taiwanese J. Math.

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Many interesting families of rapidly convergent series representations for the Riemann Zeta function ζ(2n + 1) (n ∈ N) were considered recently by various authors. In this survey-cum-expository paper, the author presents a systematic (and historical) investigation of these series representations. Relevant connections of the results presented here with several other known series representations for ζ(2n + 1) (n ∈ N) are also pointed out. In one of many computationally useful special cases presented here, it is observed that ζ(3) can be represented by means of a series which converges much faster than that in Euler's celebrated formula as well as the series used recently by Apéry in his proof of the irrationality of ζ(3). Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of this series are capable of producing an accuracy of seven decimal places.

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Section: The First Set Of Series Representationsmentioning

confidence: 96%

“…the following series representations for ζ(2n + 1) were proven recently by appealing appropriately to the series identity (2.2) in its special cases when m = 2, 3, 4, and 6 (see Srivastava [24]):…”

Section: The First Set Of Series Representationsmentioning

confidence: 99%

Some Families of Rapidly Convergent Series Representations for the Zeta Functions

Srivástava¹

2000

Taiwanese J. Math.

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“…In 1999, by using another method in reference [6], H. M. Srivastava has obtained the following results (n ∈ N):…”

Section: Basic Formulas Of Abstract Operatorsmentioning

confidence: 99%

New Properties of Fourier Series and Riemann Zeta Function

Bi,

2010

Preprint

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We establish the mapping relations between analytic functions and periodic functions using the abstract operators cos(h∂x) and sin(h∂x), including the mapping relations between power series and trigonometric series, and by using such mapping relations we obtain a general method to find the sum function of a trigonometric series. According to this method, if each coefficient of a power series is respectively equal to that of a trigonometric series, then if we know the sum function of the power series, we can obtain that of the trigonometric series, and the non-analytical points of which are also determined at the same time, thus we obtain a general method to find the sum of the Dirichlet series of integer variables, and derive several new properties of ζ(2n + 1).

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“…Nevertheless, we will derive several integral and series representations, related to these numbers and general positive numbers greater than one, involving our Sheffer's sequences of polynomials. Some rapidly convergent series for ζ(2n + 1) see, for instance, in [17].…”

Section: Riemann's Zeta-valuesmentioning

confidence: 99%