\documentclass{article}
\begin{document}
\begin{rawhtml}
# The sum of divisors of $n$

\end{rawhtml}
Let
$$
\sigma(n) = \sum_{d|n} d
$$
denote the sum of the divisors of $n$.
G.~Robin [
\begin{rawhtml}
MR 86f:11069
\end{rawhtml}] showed that the Riemann Hypothesis is
equivalent to
$$
\sigma(n) < e^\gamma n \log\log n
$$
for all $n\ge 5041$, where $\gamma$ is Euler's constant.
That inequality does not leave much to spare, for Gronwall showed
$$
\limsup_{n\to\infty} \frac{\sigma(n)}{n \log\log n} = e^\gamma ,
$$
and Robin showed unconditionally that
$$
\sigma(n) < e^\gamma n \log\log n + 0.6482 \frac{n}{\log\log n},
$$
for $n\ge 3$.
J. Lagarias [
\begin{rawhtml}
arXiv:math.NT/0008177
\end{rawhtml}] elaborated on Robin's
work and showed that the Riemann Hypothesis is equivalent to
$$
\sigma(n) < H_n + \exp(H_n)\log(H_n)
$$
for all $n\ge 2$, where $H_n$ is the harmonic number
$$
H_n=\sum_{j=1}^n \frac{1}{j} .
$$
By definition,
$$
\gamma = \lim_{n\to\infty} H_n - \log n ,
$$
so Lagarias' and Robin's inequalities are the same to leading order.
\begin{rawhtml}

Back to the
main index
for The Riemann Hypothesis.
\end{rawhtml}
\end{document}