New Criterion for the Riemann Hypothesis

EasyChair Preprint 11415, version 14

Abstract

Let $\Psi(n) = n \cdot \prod_{q \mid n} \left(1 + \frac{1}{q} \right)$ denote the Dedekind $\Psi$ function where $q \mid n$ means the prime $q$ divides $n$. Define, for $n \geq 3$; the ratio $R(n) = \frac{\Psi(n)}{n \cdot \log \log n}$ where $\log$ is the natural logarithm. Let $N_{n} = 2 \cdot \ldots \cdot q_{n}$ be the primorial of order $n$. There are several statements equivalent to the Riemann hypothesis. We prove if for all prime numbers $q_{n}$ (greater than some threshold), there exists another prime $q_{n'} > q_{n}$ such that $R(N_{n'}) \leq R(N_{n})$, then the Riemann hypothesis is true. In this note, using our criterion, we show that the Riemann hypothesis is true.

Keyphrases: Chebyshev function, Riemann hypothesis, Riemann zeta function, elementary number theory, prime numbers

@booklet{EasyChair:11415,
  author    = {Frank Vega},
  title     = {New Criterion for the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 11415},
  year      = {EasyChair, 2024}}