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

2012/11/20 by Idowu, Michael A.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1211.5033

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.

Related

• Riemann Zeta Function. An Attempt to fathom Zeta(3)
• Elegant expressions and generic formulas for the Riemann zeta function for integer arguments
• A variation of Euler's approach to values of the Riemann zeta function
• On the recurrence formula of the Euler zeta functions
• Recurrence Relations for Values of the Riemann Zeta Function in Odd Integers