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 Discrete limit theorems for the Mellin transform of the Riemann zeta-function

2008 ◽  
Vol 131 (1) ◽  
pp. 29-42 ◽  
Author(s):  
Violeta Balinskaitė ◽  
Antanas Laurinčikas
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Limit Theorems ◽  
Mellin Transform ◽  
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 two-dimensional limit discrete theorem for Mellin transforms of the Riemann zeta-function

2007 ◽  
Vol 47 ◽  
Author(s):  
Violeta Balinskaitė
Keyword(s):  
Limit Theorem ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mellin Transform ◽  
Two Dimensional ◽  
Mellin Transforms ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In the paper two-dimensional limit theorem for the modified Mellin transform of the Riemann zeta-function is obtained.

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 ?.

ZERO-FREE REGIONS OF A q-ANALOGUE OF THE COMPLETE RIEMANN ZETA FUNCTION

2011 ◽  
Vol 07 (04) ◽  
pp. 1075-1092
Author(s):  
MITSUGU MERA
Keyword(s):  
Zeta Function ◽  
Theta Function ◽  
Riemann Zeta Function ◽  
Mellin Transform ◽  
Critical Strip ◽  
Jacobi Theta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

A q-analogue of the complete Riemann zeta function presented in this paper is defined by the q-Mellin transform of the Jacobi theta function. We study zero-free regions of the q-zeta function. As a by-product, we show that the Riemann zeta function does not vanish in a sub-region of the critical strip.

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