Zeta(n) via hyperbolic functions

TL;DR

This paper introduces a novel method for computing the Riemann zeta function at 2 and extends it to values greater than 2 using hyperbolic functions, transforming the problem into single-variable integrals.

Contribution

It presents a new integral representation of the zeta function leveraging hyperbolic functions, simplifying the computation process for various n.

Findings

Derived a new integral formula for ζ(2) using hyperbolic substitution.

Extended the approach to express ζ(n) for n > 2 as single-variable integrals.

Abstract

We present here an approach to a computation of $\zeta(2)$ by changing variables in the double integral using hyperbolic trig functions. We also apply this approach to present $\zeta(n)$, when $n>2$, as a definite improper integral of single variable.