An Equivalent Statement to Nicolas' Criterion

TL;DR

This paper reformulates Nicolas' criterion for the Riemann Hypothesis, proving its equivalence and analyzing its properties using prime number functions, while also discussing its consistency with Cramer's conjecture.

Contribution

It provides a new equivalent formulation of Nicolas' criterion and analyzes its properties, connecting it with prime number theory and conjectures.

Findings

Reformulation of Nicolas' criterion as an equivalent statement.

Bounded and monotonic properties established using Chebyshev's function.

No contradiction with Cramer's conjecture found.

Abstract

Nicolas' criterion for the Riemann Hypothesis (RH) is an inequality based on primorials and the Euler totient function. The aim of this paper is to reformulate Nicolas' criterion and prove the equivalent statement. I will show that the reformulation is bounded and montonic using Chebyshev's function and results on prime numbers. I will then show this equivalent statement does not contradict Cramer's conjecture, which arises naturally when one would prove a specific sequence related to that bound is strictly decreasing.