Evaluation of Riemann Zeta function on the Line $\Re(s) = 1$ and Odd Arguments

TL;DR

This paper introduces a new simple formula for approximating the Riemann Zeta function at odd arguments with exponential convergence and evaluates its behavior on the line Re(s) = 1, comparing it with existing methods.

Contribution

It provides a novel simple approximation formula for the zeta function at odd arguments and evaluates its value on the line Re(s) = 1, enhancing computational techniques.

Findings

The new formula converges exponentially for odd arguments.

Comparison shows the formula's accuracy exceeds existing methods.

Evaluation on Re(s)=1) offers insights into zeta function behavior.

Abstract

We have looked at the evaluation of the Riemann Zeta function at odd arguments and have provided a simple formula to approximate the value with exponential convergence. We have compared it with various other formulae present in literature. We have also evaluated an expression for the zeta function on the plane $\Re(s) = 1$.