# 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} }

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.

## 87 Citations

### Robin's Inequality for Sum of Divisors Function and the Riemann Hypothesis

- Mathematics
- 2012

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

- Mathematics
- 2021

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

- Mathematics
- 2012

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

- Mathematics
- 2021

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

- Mathematics
- 2012

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

- Mathematics, Philosophy
- 2016

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

- Mathematics, Philosophy
- 2022

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

- Mathematics
- 2017

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

- Mathematics
- 2012

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…

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