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La relation entre $ζ(4n-1)$, $ζ(2p)$ et $ζ(4n-1-2p)$

Mundankulu Kabongo

Abstract

The functional relation of the Riemann zêta function provides us with neither the nature nor the expression of zêta at positive odd numbers. From the function $F(z)=\frac{z^{-2n}}{e^z-1}$, we find a functional relation involving $ζ(4n- 1)$, $ζ(2p)$ and $ζ(4n-1-2p)$. It is given by: \begin{equation} ζ(4n-1)=\frac{1}{2n-1}\sum_{p=1}^{2n-2}ζ(2p)ζ(4n-1-2p). \end{equation} $n=2, 3, 4, 5, 6, ...$ From this formula we introduce a new approach to study the nature of $ζ$ on these integers.

La relation entre $ζ(4n-1)$, $ζ(2p)$ et $ζ(4n-1-2p)$

Abstract

The functional relation of the Riemann zêta function provides us with neither the nature nor the expression of zêta at positive odd numbers. From the function , we find a functional relation involving , and . It is given by: \begin{equation} ζ(4n-1)=\frac{1}{2n-1}\sum_{p=1}^{2n-2}ζ(2p)ζ(4n-1-2p). \end{equation} From this formula we introduce a new approach to study the nature of on these integers.
Paper Structure (5 sections, 51 equations)

This paper contains 5 sections, 51 equations.

Table of Contents

  1. Introduction
  2. Démonstration de $\zeta(4n-1)$
  3. Nouvelle approche pour étudier la nature de zeta aux impairs positifs
  4. Conclusion
  5. Références

Theorems & Definitions (2)

  • remark 1
  • remark 2