# On Robin's criterion for the Riemann Hypothesis

@article{Choie2006OnRC,
title={On Robin's criterion for the Riemann Hypothesis},
author={YoungJu Choie and Nicolas Lichiardopol and Pieter Moree and P. Sol{\'e}},
journal={Journal de Theorie des Nombres de Bordeaux},
year={2006},
volume={19},
pages={357-372}
}
• Published 13 April 2006
• Mathematics, Philosophy
• Journal de Theorie des Nombres de Bordeaux
Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality (n) := P d|n d 1. As consequence we obtain that RH holds true i every natural number divisible by a fifth power > 1 satisfies Robin's inequality.

Let $\sigma ( n)$ denote the sum of divisors function. In this paper we give a simple proof of the  Robin inequality \textbf{(R)}: $\sigma ( n) (R) implies Riemann Hypothesis. ### A note on Robin's criterion for the Riemann hypothesis Define$\sigma(n):=\sum_{d \mid n} d$and$G(n):=\frac{\sigma(n)}{n \log \log n}$. Basing on some result of Robin on$G(n)$, we demonstrate that the Riemann hypothesis must be true. ### The Riemann Hypothesis and the Robin Inequality The Riemann hypothesis is one of the most important unsolved problems in the modern mathematics. The Riemann hypothesis is closely related with the distribution of prime numbers. The Robin inequality ### A new inequality for the Riemann hypothesis There have been published many research results on the Riemann hypothesis. In this paper, we first find a new inequality for the Riemann hypothesis on the basis of wellknown Robin theorem. Next, we ### An Equivalent Inequality to the Riemann Hypothesis The Riemann hypothesis is well known. The Riemann hypothesis is related with many problems of the analytical number theory. And there have been found some propositions equivalent to one. In ### The Riemann Hypothesis Robin criterion states that the Riemann Hypothesis is true if and only if the inequality σ ( n ) < e γ × n × loglog n holds for all n > 5040, where σ ( n ) is the sum-of-divisors function and γ ≈ 0 . ### Counterexample of the Riemann Hypothesis Under the assumption that the Riemann hypothesis is true, von Koch deduced the improved asymptotic formula$\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$, where$\theta(x)$is the Chebyshev ### Robin's inequality for 20-free integers. • Mathematics • 2018 In 1984, Robin showed that the Riemann Hypothesis for$\zeta$is equivalent to demonstrating$\sigma(n) 5040$. Robin's inequality has since been proven for various infinite families of power-free ### A New Sufficient Condition by Euler Function for Riemann Hypothesis The aim of this paper is to show a new sufficient condition (NSC) by the Euler function for the Riemann hypothesis and its possibility. We build the NSC for any natural numbers ≥ 2 from well-known ### Some Sufficient Conditions for the Riemann hypothesis The Riemann hypothesis (RH) is well known. In this paper we would show some sufficient conditions for the RH. The first condition is related with the sum of divisors function and another one is ## References SHOWING 1-10 OF 20 REFERENCES ### An Elementary Problem Equivalent to the Riemann Hypothesis The function a (n) = dInd is the sum-of-divisors function, so for example a (6) = 12. The number Hn is called the nth harmonic number by Knuth, Graham, and Patashnik [12, sect. 6.3], who detail ### Abundant Numbers and the Riemann Hypothesis A computational study of the successive maxima of the relative sum-of-divisors function ρ(n) := σ(n)/n, which occurs at superabundant and colossally abundant numbers and the density of these numbers is studied. ### On Highly Composite Numbers for a certain c. In fact I shall prove that if n is highly composite, then the next highly composite number is less than n+n(log y&)-C ; and the result just stated follows immediately from this. At, ### Introduction to Analytic and Probabilistic Number Theory Foreword Notation Part I. Elementary Methods: Some tools from real analysis 1. Prime numbers 2. Arithmetic functions 3. Average orders 4. Sieve methods 5. Extremal orders 6. The method of van der ### Introduction to analytic number theory This is the first volume of a two-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years. It provides an introduction ### Highly Composite Numbers by Srinivasa Ramanujan • Mathematics • 1997 In 1915, the London Mathematical Society published in its Proceedings a paper of Ramanujan entitled “Highly Composite Numbers”. But it was not the whole work on the subject, and in “The lost notebook ### Ordre maximal d'un élément du groupe$S_n\$ des permutations et «highly composite numbers»

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### Mathematical constants

• S. Finch
• Mathematics
Encyclopedia of mathematics and its applications
• 2005
UCBL-20418 This collection of mathematical data consists of two tables of decimal constants arranged according to size rather than function, a third table of integers from 1 to 1000, giving some of