The sum of divisors of $n$

Let

\begin{displaymath} \sigma(n) = \sum_{d\vert n} d \end{displaymath}

denote the sum of the divisors of $n$.

G. Robin [ MR 86f:11069] showed that the Riemann Hypothesis is equivalent to

\begin{displaymath} \sigma(n) < e^\gamma n \log\log n \end{displaymath}

for all $n\ge 5041$, where $\gamma$ is Euler's constant. That inequality does not leave much to spare, for Gronwall showed

\begin{displaymath} \limsup_{n\to\infty} \frac{\sigma(n)}{n \log\log n} = e^\gamma , \end{displaymath}

and Robin showed unconditionally that

\begin{displaymath} \sigma(n) < e^\gamma n \log\log n + 0.6482 \frac{n}{\log\log n}, \end{displaymath}

for $n\ge 3$.

J. Lagarias [ arXiv:math.NT/0008177] elaborated on Robin's work and showed that the Riemann Hypothesis is equivalent to

\begin{displaymath} \sigma(n) < H_n + \exp(H_n)\log(H_n) \end{displaymath}

for all $n\ge 2$, where $H_n$ is the harmonic number

\begin{displaymath} H_n=\sum_{j=1}^n \frac{1}{j} . \end{displaymath}

By definition,

\begin{displaymath} \gamma = \lim_{n\to\infty} H_n - \log n , \end{displaymath}

so Lagarias' and Robin's inequalities are the same to leading order.



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