Elegant expressions and generic formulas for the Riemann zeta function for integer arguments

TL;DR

The paper introduces a new, elegant formula for the Riemann zeta function at positive integers, simplifying calculations especially for small even integers and providing a straightforward proof for zeta(2).

Contribution

It presents a novel, elegant expression for the Riemann zeta function at positive integers, enabling easier computation and a simple proof for zeta(2).

Findings

New formula for zeta(s) for positive integers

Easy calculation method for small even integers

One-line proof of zeta(2) using cotangent

Abstract

A new definition for the Riemann zeta function for all positive integer number s > 1 is presented. We discover a most elegant expression and easy method for calculating the Riemann zeta function for small even integer values. Through this new reformulation we provide a one-line proof of the value of zeta(2) and demonstrate that zeta(2s) may be calculated by hand using only the cotangent function when the magnitude of the integer s is small.