Related papers: Properties of counterexample to Robin hypothesis

Let $G(n)=\sigma (n)/(n \log \log n )$. Robin made hypothesis that $G(n)<e^\gamma$ for all integer $n>5040$. If Robin hypothesis fails, there will be a least counterexample. This article collects the requirements the least counterexample…

Let $G(n)=\sigma (n)/(n \log \log n )$. Robin made hypothesis that $G(n)<e^\gamma$ for all integer $n>5040$. This article divides all colossally abundant numbers in to three disjoint subsets CA1, CA2 and CA3, and shows that Robin hypothesis…

Robin's Inequality posits $G(n)<e^{\gamma}$ for $n>5040$. Robin also showed that if the Riemann Hypothesis (RH) is false, then $G(n)>e^{\gamma}\left(1+\displaystyle\frac{c}{(\log n)^{b}}\right)$ for infinitely many values of $n$. By…

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $\sigma(n)<e^\gamma n\log\log n$ holds for every integer $n>5040$, where $\sigma(n)$ is the sum of divisors function, and $\gamma$ is the…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

Let $\sigma(n)$ denotes the sum of divisors function of a positive integer $n$. Robin proved that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma}n \log \log n$ holds for every positive integer $n \geq…

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality $\sigma(n):=\sum_{p|n}p<e^{\gamma} n \log \log n$ is satisfied for $n > 5040$, where $\gamma$ denotes the Euler-Mascheroni constant. We show…

Robin's theorem is one of the ingenious reformulation of the Riemann hypothesis (RH). It states that the RH is true if and only if $\sigma(n)<e^\gamma n\log\log n$ for all $n>5040$ where $\sigma(n)$ is the sum of divisors of $n$ and…

In 1984, Robin showed that the Riemann Hypothesis for $\zeta$ is equivalent to demonstrating $\sigma(n) < e^\gamma n \log \log n$ for all $n > 5040$. Robin's inequality has since been proven for various infinite families of power-free…

Define $s (n) := n^{- 1} \sigma (n)$ ($\sigma (n):=\sum_{d|n}d )$ and $\omega(n)$ is the number of prime divisors of $n$. One of the properties of $s$ plays a central role: $s (p^a) > s (q^b)$ if $p < q$ are prime numbers, with no special…

In this paper we give a variant of the Robin inequality which states that $\frac{\sigma(n)}{n} \leq \frac{e^\gamma}{2} \log\log n + \frac{0.7398\cdots}{\log\log n}$ for any odd integer $n \geq 3$.

For n>1, let G(n)=\sigma(n)/(n log log n), where \sigma(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime…

Robin's criterion states that the Riemann hypothesis is equivalent to $\sigma(n) < e^\gamma n \log\log n$ for all integers $n \geq 5041$, where $\sigma(n)$ is the sum of divisors of $n$ and $\gamma$ is the Euler-Mascheroni constant. We…

The conjectured Robin inequality for an integer $n>7!$ is $\sigma(n)<e^\gamma n \log \log n,$ where $\gamma$ denotes Euler constant, and $\sigma(n)=\sum_{d | n} d $. Robin proved that this conjecture is equivalent to Riemann hypothesis…

As essential condition for the validy of Robin's Theorem as a precondition for the proof of the Riemann hypothesis, we show that the minimum of the function $F={\rm e}^{\gamma}\,\ln(\ln\,n)-\sigma(n)/n$ is found to be positive. Therefore,…

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality sum_{d|n}d<e^{gamma}n loglog n is satisfied for n>=5041, where gamma denotes the Euler(-Mascheroni) constant. We show by elementary methods…

The Robin criterion states that the Riemann hypothesis is equivalent to the inequality $\sigma(n) < e^\gamma n \log \log n$ for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, and $\gamma$ is the Euler--Mascheroni constant.…

We consider the function $G(n)=\frac{\sigma(n)}{n\log\log n}$ (where $\sigma(n)=\sum_{d|n}d$) and set an imposed condition on its argument $n$, the fulfillment of which is sufficient for the existence of a prime $p$, at which $G(np)>G(n)$.…

There are many formulations of problems that have been proven to be equivalent to the Riemann Hypothesis in modern mathematics. In this paper we look at the formulation of an inequality derived by Robin in 1984 that proves the Riemann…

Recall that an integer is $t-$free iff it is not divisible by $p^t$ for some prime $p.$ We give a method to check Robin inequality $\sigma(n) < e^\gamma n\log\log n,$ for $t-$free integers $n$ and apply it for $t=6,7.$ We introduce…