A closed-form expression for zeta(2n+1) reveals a self-recursive function

TL;DR

This paper presents a closed-form expression for the Riemann zeta function at odd integers, revealing its self-recursive nature through a recurrence relation involving known mathematical constants and functions.

Contribution

It introduces a novel closed-form formula for zeta(2n+1) based on Dirichlet beta function, Euler numbers, and pi, demonstrating the self-recursive property at odd integers.

Findings

Zeta(2n+1) can be expressed in closed form involving known constants.

The Riemann zeta function at odd integers exhibits a self-recursive recurrence relation.

First demonstration of the self-recursive nature of zeta at odd integers.

Abstract

Euler discovered a formula for expressing the value of the Riemann zeta function for all even positive integer arguments. A closed-form expression for the Riemann zeta function for all odd integer arguments, based on the values of the Dirichlet beta function, euler numbers and pi, reveals a new evidence about the self-recursive nature of Riemann zeta function at odd integers. We demonstrate for the first time that the Riemann zeta function at odd integers always produces a recurrence relation that is self-recursive.