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 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 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.

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 Many odd zeta values are irrational

2019 ◽  
Vol 155 (5) ◽  
pp. 938-952 ◽  
Author(s):  
Stéphane Fischler ◽  
Johannes Sprang ◽  
Wadim Zudilin
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Real Number ◽  
Riemann Zeta Function ◽  
Positive Real Number ◽  
Positive Real ◽  
Linear Forms ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta

Building upon ideas of the second and third authors, we prove that at least$2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$values of the Riemann zeta function at odd integers between 3 and$s$are irrational, where$\unicode[STIX]{x1D700}$is any positive real number and$s$is large enough in terms of$\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of$\log s$; it improves on the bound$(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

On functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function

2007 ◽  
Vol 142 (3) ◽  
pp. 395-405 ◽  
Author(s):  
HIROFUMI TSUMURA
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Zeta Functions ◽  
Analytic Functional ◽  
Functional Relations ◽  
Double Zeta ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Positive Integers

AbstractIn this paper, we give certain analytic functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell–Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950's.

Zeta Functions and the Log Behaviour of Combinatorial Sequences

2015 ◽  
Vol 58 (3) ◽  
pp. 637-651 ◽  
Author(s):  
William Y. C. Chen ◽  
Jeremy J. F. Guo ◽  
Larry X. W. Wang
Keyword(s):  
Probability Theory ◽  
Zeta Function ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Bernoulli Number ◽  
Hölder’S Inequality ◽  
Zeta Functions ◽  
Bell Number ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

AbstractIn this paper, we use the Riemann zeta functionζ(x) and the Bessel zeta functionζμ(x) to study the log behaviour of combinatorial sequences. We prove thatζ(x) is log-convex forx> 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n≥ 1 is log-convex, whereBnis thenth Bernoulli number. We introduce the functionθ(x) = (2ζ(x)Γ(x + 1))1/x, whereΓ(x)is the gamma function, and we show that logθ(x) is strictly increasing forx≥ 6. This confirms a conjecture of Sun stating that the sequenceis strictly increasing. Amdeberhanet al. defined the numbersan(μ)= 22n+1(n+ 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence{an(μ)}n≥1is log-convex forμ= 0 andμ= 1. By proving thatζμ(x)is log-convex forx >1 andμ >-1, we show that the sequence{an(≥)}n>1 is log-convex for anyμ >- 1. We introduce another functionθμ,(x)involvingζμ(x)and the gamma functionΓ(x)and we show that logθμ(x)is strictly increasing forx >8e(μ+ 2)2. This implies thatBased on Dobinski’s formula, we prove thatwhereBnis thenth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property ofand Holder’s inequality in probability theory.

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