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<h1 align=center> The sum of divisors of $n$
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Let 
$$
\sigma(n) = \sum_{d|n} d
$$
denote the sum of the divisors of $n$.

G.~Robin [
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<a href="http://www.ams.org/msnmain?fn=130&s1=86f:11069&pg1=MR">MR 86f:11069</a>
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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 [
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<a href="http://www.arxiv.org/abs/math/0008177">arXiv:math.NT/0008177</a>
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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.

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<p> <br>
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<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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