On Apery's Constant and Catalan's Constant

TL;DR

This paper explores the representation of Riemann's Zeta function at odd integers as sums involving powers of pi with rational coefficients, deriving specific values for Apery's and Catalan's constants.

Contribution

It introduces a novel infinite summation representation for the zeta function at odd integers and derives explicit formulas for Apery's and Catalan's constants.

Findings

Derived new formulas for Apery's and Catalan's constants.

Expressed zeta function at odd integers as sums of powers of pi with rational coefficients.

Provided insights into the structure of these mathematical constants.

Abstract

In this paper, Riemann's Zeta function with odd positive integer argument is represented as an infinite summation of integer powers of $\pi$ with rational coefficients. Specific values for Apery's Constant and Catalan's Constant are then derived.