On Robin’s criterion for the Riemann hypothesis

2008/12/02 by YoungJu Choie, Nicolas Lichiardopol, Pieter Moree +1 · 2 citations
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Advanced Algebra and Geometry #Mathematics #Combinatorics #Riemann hypothesis #Humanities #Calculus (dental) #Pure mathematics #Philosophy

paper · pdf · doi:10.5802/jtnb.591

openalex publication_date 2008/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

Abstract

Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>|</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mi>d</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>γ</mml:mi> </mml:msup> <mml:mi>n</mml:mi> <mml:mo form="prefix">log</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> is satisfied for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5041</mml:mn> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> denotes the Euler(-Mascheroni) constant. We show by elementary methods that if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>37</mml:mn> </mml:mrow> </mml:math> does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> must be divisible by a fifth power <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . As consequence we obtain that RH holds true iff every natural number divisible by a fifth power <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> satisfies Robin’s inequality.

Citations

• Highly Composite Numbers by Srinivasa Ramanujan
• Approximate formulas for some functions of prime numbers
• Abundant Numbers and the Riemann Hypothesis
• An Elementary Problem Equivalent to the Riemann Hypothesis
• Highly Composite Numbers

Cited by

• Analogues of the Robin-Lagarias Criteria for the Riemann Hypothesis
• Divisor function [wikipedia]

Related

• Analogues of the Robin-Lagarias Criteria for the Riemann Hypothesis
• A family of analogues to the Robin criterion
• Riemann's hypothesis via Robin's Theorem
• Riemann hypothesis equivalences, Robin inequality, Lagarias criterion, and Riemann hypothesis
• Properties of Prime Products and Robin's Inequality