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A New Representation of the Riemann Zeta Function $ζ(s)$

S. C. Woon

TL;DR

This work develops a novel representation of the Riemann zeta function $\zeta(s)$ in terms of a nested Bernoulli-number series, extending the classical $\zeta(2n)$–Bernoulli relation to a general complex parameter. It introduces a binary-tree framework with operators $O_L$ and $O_R$ to generate Bernoulli numbers and extends this to a Bernoulli function $B(s)$, addressing the sign ambiguity of $B_1$ through a consistent reformulation. By linking $\zeta(1-s)=-B(s)/s$ to the zeta functional equation, the paper derives the main representation of $\zeta(s)$ as a limit involving the tree-generated Bernoulli structure. The approach provides an analytic-structural perspective on $\zeta(s)$ via operator-driven Bernoulli generation and a consistent Bernoulli sign convention, offering a new avenue for studying zeta-values and Bernoulli identities.

Abstract

A generalization of a well-known relation between the Riemann zeta function $ζ(s)$ and Bernoulli numbers $B_n$ is obtained. The formula is a new representation of the Riemann zeta function in terms of a nested series of Bernoulli numbers.

A New Representation of the Riemann Zeta Function $ζ(s)$

TL;DR

This work develops a novel representation of the Riemann zeta function in terms of a nested Bernoulli-number series, extending the classical –Bernoulli relation to a general complex parameter. It introduces a binary-tree framework with operators and to generate Bernoulli numbers and extends this to a Bernoulli function , addressing the sign ambiguity of through a consistent reformulation. By linking to the zeta functional equation, the paper derives the main representation of as a limit involving the tree-generated Bernoulli structure. The approach provides an analytic-structural perspective on via operator-driven Bernoulli generation and a consistent Bernoulli sign convention, offering a new avenue for studying zeta-values and Bernoulli identities.

Abstract

A generalization of a well-known relation between the Riemann zeta function and Bernoulli numbers is obtained. The formula is a new representation of the Riemann zeta function in terms of a nested series of Bernoulli numbers.

Paper Structure

This paper contains 5 sections, 4 theorems, 37 equations, 2 figures.

Table of Contents

  1. A New Representation of the Riemann Zeta Function
  2. A Binary Tree for Generating Bernoulli Numbers
  3. The Tree-Generating Operator and Bernoulli Function
  4. Fixing of the Arbitrary Sign Convention of $B_1$
  5. Proof of the New Representation of $\zeta(s)$

Key Result

Theorem 1

for $\hbox{\sf Re}(s)>(1/w)$ where $s\in\bf\sf C, \;w\in\bf\sf R, \;w>0$, the notation of binomial coefficient is extended such that $B_m$ are the Bernoulli numbers with $B_1 = 1/2$, and the limit only needs to be taken when $s\in\{1,3,5,\dots\}$ for which the denominator $\cos\,(\pi s/2)$ is $0$.

Figures (2)

  • Figure 1: The binary tree that generates Bernoulli numbers.
  • Figure 2: The curve $B(s)$ runs through the points of all $(n,B_n)$ except $(1,B_1)$.

Theorems & Definitions (7)

  • Theorem 1
  • Definition 1
  • Definition 2: The $B_n$-Generating Tree
  • Theorem 2
  • Lemma 1
  • Theorem 3: Bernoulli Function
  • Definition 3: Redefinition of Bernoulli Numbers