Corpus ID: 119132402

Some infinite series involving the Riemann zeta function

@article{Connon2012SomeIS,
  title={Some infinite series involving the Riemann zeta function},
  author={Donal F. Connon},
  journal={arXiv: Classical Analysis and ODEs},
  year={2012}
}
This paper considers some infinite series involving the Riemann zeta function. Some examples are set out below 
The Euler–Riemann zeta function in some series formulae and its values at odd integer points
Abstract The paper presents formulae for certain series involving the Riemann zeta function. These formulae are generalizations, in a natural way, of well known formulae, originating from LeonhardExpand
Evaluation of log-tangent integrals by series involving
ABSTRACT In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on may be determined (or approximated) by an infinite (orExpand
N ov 2 01 6 Evaluation of Log-tangent Integrals by series involving ζ ( 2 n + 1 ) BY
In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on [ 0, π 2 ] may be determined by a finite or infinite sum involving theExpand
Euler-type formulas
Finding the exact value that � ∞=1 1 n3 converges to is one of the most notorious problems that did not even yield to Euler. A less difficult problem to consider is to find a representation of ζ(3)Expand
On the distribution of the nontrivial zeros of the Riemann zeta function
This thesis is a bunch of recent results which treat, in view of its strong connexion, three principle open problems in the theory of the Riemann zeta function: algebraic properties of the zetaExpand
“Sixth root of unity” and Feynman diagrams: hypergeometric function approach point of view
We briefly discuss the transcendental constants generated through the e expansion of generalized hypergeometric functions and their interrelation with the “sixth root of unity.”
Some properties and results involving the zeta and associated functions
In this research-cum-expository article, we aim at presenting a systematic account of some recent developments involving the Riemann Zeta function ζ(s), the Hurwitz (or generalized) Zeta functionExpand
New Properties of Fourier Series and Riemann Zeta Function
We establish the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relations betweenExpand
Riemann, Hurwitz and Hurwitz-Lerch Zeta Functions and Associated Series and Integrals
The main object of this article is to present a survey-cum-expository account of some recent developments involving the Riemann Zeta function \(\zeta (s)\), the Hurwitz (or generalized) Zeta functionExpand

References

SHOWING 1-10 OF 59 REFERENCES
Closed-form evaluations of definite integrals and associated infinite series involving the Riemann zeta function
TLDR
A class of definite integrals are evaluated using the Cauchy residue theorem and used to obtain closed-form expressions for several infinite series associated with the Riemann zeta function. Expand
Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume VI
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, someExpand
A class of logarithmic integrals
A class of de nite integrals involving cyclotomic polynomials and nested logarithms is considered. The results are given in terms of derivatives of the Hurwitz Zeta function. Some special cases forExpand
Series Associated with the Zeta and Related Functions
Preface. Acknowledgements. 1. Introduction and Preliminaries. 2. The Zeta and Related Functions. 3. Series Involving Zeta Functions. 4. Evaluations and Series Representations. 5. Determinants of theExpand
Euler’s integrals and multiple sine functions
We show that Euler's famous integrals whose integrands contain the logarithm of the sine function are expressed via multiple sine functions.
Some Representations of pi
In this paper we consider a particular integral from which we may develop identities for pi.
Integer Powers of Arcsin
TLDR
New simple nested-sum representations for powers of the arcsin function are given, which makes connections to finite binomial sums and polylogarithms. Expand
On Some Integrals Involving the Hurwitz Zeta Function: Part 2
AbstractWe establish a series of indefinite integral formulae involving the Hurwitz zeta function and other elementary and special functions related to it, such as the Bernoulli polynomials, lnExpand
Contributions to the Theory of the Barnes Function
This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for high-precision computation of the Barnes gamma functionExpand
Some interesting series arising from the power series expansion of (sin-1x)q
  • Habib Muzaffar
  • Computer Science, Mathematics
  • Int. J. Math. Math. Sci.
  • 2005
Starting from the power series expansions of ( sin - 1 x ) q , for 1 ≤ p ≤ 4 , formulae are obtained for the sum of several infinite series. Some of these evaluations involve ζ ( 3 ) .
...
1
2
3
4
5
...