scholarly journals Some generating functions of the Riemann zeta function

2019 ◽  
Vol 118 ◽  
pp. 107-111
Author(s):  
Takumi Noda
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Generating Functions ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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 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 Generating Function Identities for $\zeta(2n+2), \zeta(2n+3)$ via the WZ Method

10.37236/759 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Kh. Hessami Pilehrood ◽  
T. Hessami Pilehrood
Keyword(s):  
Zeta Function ◽  
Generating Function ◽  
Riemann Zeta Function ◽  
Generating Functions ◽  
Convergent Series ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Wz Method

Using WZ-pairs we present simpler proofs of Koecher, Leshchiner and Bailey-Borwein-Bradley's identities for generating functions of the sequences $\{\zeta(2n+2)\}_{n\ge 0}$ and $\{\zeta(2n+3)\}_{n\ge 0}.$ By the same method, we give several new representations for these generating functions yielding faster convergent series for values of 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 .

scholarly journals On The Tails of the Exponential Series

1994 ◽  
Vol 37 (2) ◽  
pp. 278-286 ◽  
Author(s):  
C. Yalçin Yildirim
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Partial Sums ◽  
Maclaurin Series ◽  
Exponential Series ◽  
Asymptotic Estimation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

AbstractA relation between the zeros of the partial sums and the zeros of the corresponding tails of the Maclaurin series for ez is established. This allows an asymptotic estimation of a quantity which came up in the theory of the Riemann zeta-function. Some new properties of the tails of ez are also provided.

Sign in / Sign up

Export Citation Format

Share Document