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 ?.

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 General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function

2021 ◽  
Vol 55 (2) ◽  
pp. 115-123
Author(s):  
R. Frontczak ◽  
T. Goy
Keyword(s):  
Zeta Function ◽  
Infinite Series ◽  
Riemann Zeta Function ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

The purpose of this paper is to present closed forms for various types of infinite seriesinvolving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments.To prove our results, we will apply some conventional arguments and combine the Binet formulasfor these sequences with generating functions involving the Riemann zeta function and some known series evaluations.Among the results derived in this paper, we will establish that $\displaystyle\sum_{k=1}^\infty (\zeta(2k+1)-1) F_{2k} = \frac{1}{2},\quad\sum_{k=1}^\infty (\zeta(2k+1)-1) \frac{L_{2k+1}}{2k+1} = 1-\gamma,$ where $\gamma$ is the familiar Euler-Mascheroni constant.

scholarly journals A generalized Davenport expansion

2021 ◽  
pp. 1-5
Author(s):  
Alexander E. Patkowski
Keyword(s):  
Zeta Function ◽  
Infinite Series ◽  
Riemann Zeta Function ◽  
Fourier Expansion ◽  
Mellin Transform ◽  
Fractional Part ◽  
Arithmetic Functions ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.

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.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

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