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.

scholarly journals Ramanujan's formula for the logarithmic derivative of the gamma function

1996 ◽  
Vol 120 (3) ◽  
pp. 391-401
Author(s):  
David Bradley
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Logarithmic Derivative ◽  
Euler’S Constant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Ramanujan’S Formula ◽  
Positive Integers ◽  
Euler's Constant

AbstractWe prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks [5]. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant γ.

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.

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 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 On multiple zeta values of even arguments

2017 ◽  
Vol 13 (03) ◽  
pp. 705-716 ◽  
Author(s):  
Michael E. Hoffman
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Generating Functions ◽  
Bernoulli Number ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta

For [Formula: see text], let [Formula: see text] be the sum of all multiple zeta values with even arguments whose weight is [Formula: see text] and whose depth is [Formula: see text]. Of course [Formula: see text] is the value [Formula: see text] of the Riemann zeta function at [Formula: see text], and it is well known that [Formula: see text]. Recently Shen and Cai gave formulas for [Formula: see text] and [Formula: see text] in terms of [Formula: see text] and [Formula: see text]. We give two formulas for [Formula: see text], both valid for arbitrary [Formula: see text], one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers [Formula: see text] and for the analogous numbers [Formula: see text] defined using multiple zeta-star values of even arguments.

scholarly journals Aq-analog of Euler's decomposition formula for the double zeta function

2005 ◽  
Vol 2005 (21) ◽  
pp. 3453-3458 ◽  
Author(s):  
David M. Bradley
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Binomial Coefficients ◽  
Euler’S Formula ◽  
Double Zeta ◽  
Decomposition Formula ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Finite Sum

The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Euler's results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of double zeta values involving binomial coefficients. Here, we establish aq-analog of Euler's decomposition formula. More specifically, we show that Euler's decomposition formula can be extended to what might be referred to as a “doubleq-zeta function” in such a way that Euler's formula is recovered in the limit asqtends to 1.

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

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