Closed-form evaluations of definite integrals and associated infinite series involving the Riemann zeta function

2006 ◽  
Vol 83 (5-6) ◽  
pp. 461-472 ◽  
Author(s):  
Young Joon Cho ◽  
Myungho Jung ◽  
Junesang Choi ◽  
H. M. Srivastava
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Infinite Series ◽  
Riemann Zeta Function ◽  
Definite Integrals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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 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 ?.

scholarly journals On the values of the Riemann zeta-function at rational arguments

2001 ◽  
Vol Volume 24 ◽  
Author(s):  
S Kanemitsu ◽  
Y Tanigawa ◽  
M Yoshimoto
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Companion Paper ◽  
Hurwitz Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Ramanujan’S Formula ◽  
Form Evaluation ◽  
Rational Arguments

International audience In a companion paper, ``On multi Hurwitz-zeta function values at rational arguments, Acta Arith. {\bf 107} (2003), 45-67'', we obtained a closed form evaluation of Ramanujan's type of the values of the (multiple) Hurwitz zeta-function at rational arguments (with denominator even and numerator odd), which was in turn a vast generalization of D. Klusch's and M. Katsurada's generalization of Ramanujan's formula. In this paper we shall continue our pursuit, specializing to the Riemann zeta-function, and obtain a closed form evaluation thereof at all rational arguments, with no restriction to the form of the rationals, in the critical strip. This is a complete generalization of the results of the aforementioned two authors. We shall obtain as a byproduct some curious identities among the Riemann zeta-values.

scholarly journals Shifted quadratic Zeta series

2004 ◽  
Vol 2004 (67) ◽  
pp. 3631-3652
Author(s):  
Anthony Sofo
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Point Of View ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is well known that the Riemann Zeta functionς(p)=∑n=1∞1/npcan be represented in closed form forpan even integer. We will define a shifted quadratic Zeta series as∑n=1∞1/(4n2−α2)p. In this paper, we will determine closed-form representations of shifted quadratic Zeta series from a recursion point of view using the Riemann Zeta function. We will also determine closed-form representations of alternating sign shifted quadratic Zeta series.

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