Riemann Zeta Function. An Attempt to fathom Zeta(3)

TL;DR

This paper explores the special value Zeta(3), deriving a new formula involving well-known constants and a rapidly converging series, and discusses the broader case of odd integer arguments of the Riemann zeta function.

Contribution

It presents a novel formula for Zeta(3) that includes a rapidly converging series with rational numbers and powers of Pi, advancing understanding of odd zeta values.

Findings

Derived a new formula for Zeta(3) involving known constants and a convergent series.

Partial success in expressing the series with common constants.

Extended the analysis to general odd positive integers of the zeta function.

Abstract

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The present paper discusses in particular the case of Zeta(3). A formula for Zeta(3) is obtained which in addition to a number of well known constants includes a rapidly converging infinite series, of which each term contains rational numbers and an even power of Pi. An attempt to convert this series into a finite number of terms containing commonly known constants is met with only partial success. The general case for zeta(2n+1) is also worked out.