Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I

Abstract

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, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

This revised paper contains some corrections and some additional material

References in corpus (28)

• Dilogarithm identities
• New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
• Algebraic Relations Between Harmonic Sums and Associated Quantities
• Analytic two-loop results for selfenergy- and vertex-type diagrams with one non-zero mass
• Analytic Continuation of Mellin Transforms up to two-loop Order
• Quasi-symmetric functions and mod p multiple harmonic sums
• Six Gluon Open Superstring Disk Amplitude, Multiple Hypergeometric Series and Euler-Zagier Sums
• Riemann Zeta Function
• A class of series acceleration formulae for Catalan's constant
• Toward verification of the Riemann hypothesis: Application of the Li criterion
• The generalized-Euler-constant function $\gamma (z)$ and a generalization of Somos's quadratic recurrence constant
• Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces and the Riemann zeta function
• Contributions to the Theory of the Barnes Function
• Special Values of Multiple Polylogarithms
• Large-order Perturbation Theory and de Sitter/Anti de Sitter Effective Actions
• A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications
• Theta and Riemann xi function representations from harmonic oscillator eigensolutions
• Effective method of computing Li's coefficients and their properties
• A Hierarchical Array of Integrable Models
• An elementary proof of Apery's theorem
• Massive-Scalar Effective Actions on Anti-de Sitter Spacetime
• An Infinite Product for e^gamma via Hypergeometric Formulas for Euler's Constant, gamma
• An Explicit Formula Relating Stieltjes Constants and Li's Numbers
• Polygamma theory, the Li/Keiper constants, and validity of the Riemann Hypothesis
• Ramanujan's Approximation to the nth Partial Sum of the Harmonic Series
• More Series related to the Euler Series
• Finding Exact Values For Infinite Sums
• Torus Knot complements: A natural series for the natural logarithm

Cited by in corpus (29)

• Expansions of generalized Euler's constants into the series of polynomials in ${\pi }^{-2}$ and into the formal enveloping series with rational coefficients only
• Some applications of the Stieltjes constants
• On an integral involving the digamma function
• The difference between two Stieltjes constants
• Various applications of the (exponential) complete Bell polynomials
• New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions
• Euler-Hurwitz series and non-linear Euler sums
• Some applications of the Dirichlet integrals to the summation of series and the evaluation of integrals involving the Riemann zeta function
• Some possible approaches to the Riemann Hypothesis via the Li/Keiper constants
• Fourier series representations of the logarithms of the Euler gamma function and the Barnes multiple gamma functions
• A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants
• Some infinite series involving the Riemann zeta function
• Some trigonometric integrals involving the log gamma and the digamma function
• A New Operator Theory of Linear Partial Differential Equations
• Yet Another Table of Integrals
• Evaluation of some simple Euler-type series
• Elementary evaluations of some Euler sums
• Determination of the Stieltjes constants at rational arguments
• Some integrals and series involving the Stieltjes constants
• A formula connecting the Bernoulli numbers with the Stieltjes constants
• Some integrals involving the Stieltjes constants
• Determination of some generalised Euler sums involving the digamma function
• An introduction to the Barnes double gamma function with an application to an integral involving the cotangent function
• Some applications of the sine and cosine integrals
• Some integrals involving the Stieltjes constants:Part II
• Euler sums and a prime numbers conundrum
• An introduction to the Bernoulli function
• Closed-form formula of Riemann zeta function and eta function for all non-zero given complex numbers via sums of powers of complex functions to disprove Riemann Hypothesis
• A Ramanujan enigma involving the first Stieltjes constant