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

TL;DR

This paper introduces new identities involving the Riemann zeta function, binomial coefficients, and harmonic numbers, derived through elementary analysis, serving as an expository foundation for a series of related works.

Contribution

It presents new formulas related to the Riemann zeta function derived via elementary methods, expanding the analytical toolkit for studying these functions.

Findings

Some identities are believed to be new.

Most formulas are derived using elementary analysis.

The paper serves as an expository introduction for a series.

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.