Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ordinality and Riemann Hypothesis I

Young Deuk Kim

TL;DR

This paper proposes a novel sufficient condition for the Riemann hypothesis based on a special ordering of $Q$, the set of finite products of distinct odd primes. By relating zeros of $\zeta$ in the critical strip to vanishing sums of an alternating Dirichlet series via the Dirichlet eta function, it constructs a framework of dyadic sequences and weighted sums. The core result states that if there exists an ordering of $Q$ for which certain double-limit expressions commute, then the Riemann hypothesis holds; otherwise, the zero off the critical line would produce a contradiction. This approach reframes RH as a combinatorial-analytic condition on prime-product orderings, offering a new theoretical avenue though it hinges on the existence of such an ordering to be proven.

Abstract

We present a sufficient condition for the Riemann hypothesis. This condition is the existence of a special ordering on the set of finite products of distinct odd primes.

Ordinality and Riemann Hypothesis I

TL;DR

This paper proposes a novel sufficient condition for the Riemann hypothesis based on a special ordering of , the set of finite products of distinct odd primes. By relating zeros of in the critical strip to vanishing sums of an alternating Dirichlet series via the Dirichlet eta function, it constructs a framework of dyadic sequences and weighted sums. The core result states that if there exists an ordering of for which certain double-limit expressions commute, then the Riemann hypothesis holds; otherwise, the zero off the critical line would produce a contradiction. This approach reframes RH as a combinatorial-analytic condition on prime-product orderings, offering a new theoretical avenue though it hinges on the existence of such an ordering to be proven.

Abstract

We present a sufficient condition for the Riemann hypothesis. This condition is the existence of a special ordering on the set of finite products of distinct odd primes.
Paper Structure (3 sections, 12 theorems, 54 equations)

This paper contains 3 sections, 12 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminary Lemmas and Theorems
  3. The Sufficient Condition for the Riemann Hypothesis

Key Result

Theorem 1.1

The zeta function has a meromorphic continuation into the entire complex plane, whose only singularity is a simple pole at $z=1$.

Theorems & Definitions (26)

  • Theorem 1.1: SS
  • Theorem 1.2: RJ,SS
  • Theorem 2.1: broughan
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 16 more