A simple series representation for Apéry's constant

2013 ◽  
Vol 97 (540) ◽  
pp. 455-460 ◽  
Author(s):  
John Melville
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Quantum Electrodynamics ◽  
Series Representation ◽  
Gyromagnetic Ratio ◽  
French Mathematician ◽  
Series Representations ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

Apéry's constant is the value of ζ (3) where ζ is the Riemann zeta function. ThusThis constant arises in certain mathematical and physical contexts (in physics for example ζ (3) arises naturally in the computation of the electron's gyromagnetic ratio using quantum electrodynamics) and has attracted a great deal of interest, not least the fact that it was proved to be irrational by the French mathematician Roger é and named after him. See [1,2].Numerous series representations have been obtained for ζ (3) many of which are rather complicated [3]. é used one such series in his irrationality proof. It is not known whether ζ (3) is transcendental, a question whose resolution might be helped by a study of an appropriate series representation of ζ (3).

scholarly journals Globally convergent series for the Riemann zeta function and the Dirichlet beta function

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Beta Function ◽  
Series Representation ◽  
Convergent Series ◽  
Globally Convergent ◽  
Series Representations ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Dirichlet Beta Function

We derive the following globally convergent series for the Riemann zeta function and the Dirichlet beta function$$\zeta(s)=\frac{1}{2^{s}-2}\sum_{k=0}^{\infty}\frac{1}{2^{2k+1}}\binom{2k+1}{k+1}\sum_{m=0}^{k}\binom{k}{m}\frac{(-1)^{m}}{(m+1)^{s}} \qquad \mbox{(where $s \neq 1+\frac{2\pi i n}{\ln 2}$)},$$$$\beta(s)=\frac{1}{4^{s}}\sum_{k=0}^{\infty}\frac{1}{(k+1)!}\left(\left(\frac{3}{4}\right)^{(k+1)}-\left(\frac{1}{4}\right)^{(k+1)}\right)\sum_{m=0}^{k}\binom{k}{m}\frac{(-1)^{m}}{(m+1)^{s}}$$using a globally convergent series for the polylogarithm function, and integrals representing the Riemann zeta function and the Dirichlet beta function. To the best of our knowledge, these series representations are new. Additionally, we give another proof of Hasse's series representation for the Riemann zeta function.

On the Riemann zeta-function

1932 ◽  
Vol 28 (3) ◽  
pp. 273-274 ◽  
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Absolute Constant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

It was proved by Littlewood that, for every large positive T, ζ (s) has a zero β + iγ satisfyingwhere A is an absolute constant.

scholarly journals Generalised Dirichlet series and Hecke's functional equation

1967 ◽  
Vol 15 (4) ◽  
pp. 309-313 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
General Class ◽  
The Riemann Zeta Function ◽  
Entire Complex ◽  
Riemann Zeta ◽  
Image Position

The generalised zeta-function ζ(s, α) is defined bywhere α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

scholarly journals Generalised Euler constants

1978 ◽  
Vol 21 (1) ◽  
pp. 25-32 ◽  
Author(s):  
J. Knopfmacher
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Laurent Expansion ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

Let the Laurent expansion of the Riemann zeta function ξ(s) about s=1 be written in the formIt has been discovered independently by many authors that, in terms of this notation, the coefficient

scholarly journals RATIONAL APPROXIMATIONS TO VALUES OF BELL POLYNOMIALS AT POINTS INVOLVING EULER’S CONSTANT AND ZETA VALUES

2012 ◽  
Vol 92 (1) ◽  
pp. 71-98
Author(s):  
KH. HESSAMI PILEHROOD ◽  
T. HESSAMI PILEHROOD
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Bell Polynomials ◽  
Rational Approximations ◽  
Euler’S Constant ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Image Position ◽  
Euler's Constant

AbstractIn this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

On Values of the Riemann Zeta Function at Integral Arguments

1991 ◽  
Vol 34 (1) ◽  
pp. 60-66 ◽  
Author(s):  
John A. Ewell
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Nonnegative Integer ◽  
Rational Numbers ◽  
Multiple Series ◽  
Special Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

AbstractFor each nonnegative integer r, is represented by a multiple series which is expressed in terms of rational numbers and the special values of the zeta function Thus, the set serves as a kind of basis for expressing all of the values

Two notes on the Riemann Zeta-function

1924 ◽  
Vol 22 (3) ◽  
pp. 234-242 ◽  
Author(s):  
J. E. Little-wood
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Prime Power ◽  
Arithmetic Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

Let Λ (n) be the arithmetic function usually so denoted, which is zero unless n is a prime power pm (m ≥ 1), when it is log p. We write as usualandwhere the dash denotes that if x is an integer the last term Λ (x) of the sum is to be taken with a factor ½. We wrute further

scholarly journals The Riemann zeta function and classes of infinite series

2017 ◽  
Vol 11 (2) ◽  
pp. 386-398 ◽  
Author(s):  
Horst Alzer ◽  
Junesang Choi
Keyword(s):  
Zeta Function ◽  
Infinite Series ◽  
Riemann Zeta Function ◽  
Catalan Constant ◽  
Series Representations ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Mathematical Constants

We present one-parameter series representations for the following series involving the Riemann zeta function ??n=3 n odd ?(n)/n sn and ??n=2 n even ?(n) n sn and we apply our results to obtain new representations for some mathematical constants such as the Euler (or Euler-Mascheroni) constant, the Catalan constant, log 2, ?(3) and ?.

Summability by Dirichlet convolutions

1967 ◽  
Vol 63 (2) ◽  
pp. 393-400 ◽  
Author(s):  
S. L. Segal
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Riemann Zeta Function ◽  
Prime Number Theorem ◽  
Summation Method ◽  
Cesàro Means ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position ◽  
Equivalent Method

Ingham (3) discusses the following summation method:A series ∑an will be said to be summable to s ifwhere, as usual, [x] indicates the greatest integer ≤ x. (An equivalent method was introduced somewhat earlier by Wintner (8), but the notation (I) for the above method and the attachment to Ingham's name seem to have become usual following [(1), Appendix IV].) The method (I) is intimately connected with the prime number theorem and the fact that the Riemann zeta-function ζ(s) has no zeros on the line σ = 1. Ingham proved, among other results, that (I) is not comparable with convergence but, nevertheless, for every δ > 0, (I) ⇒ (C, δ) and for every δ, 0 < δ < 1, (C, −δ) ⇒ (I), where the (C, k) are Cesàro means of order k.

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