Editing Divisor function (section)

==Growth rate<!--linked from 'Thomas Hakon Grönwall'-->==
In [[Big O notation#Little-o notation|little-o notation]], the divisor function satisfies the inequality:{{sfnp|Apostol|1976|p=296}}{{sfnp|Hardy|Wright|2008|pp=342-347|loc=§18.1}}
:<math>\mbox{for all }\varepsilon>0,\quad d(n)=o(n^\varepsilon).</math>
More precisely, [[Severin Wigert]] showed that:{{sfnp|Hardy|Wright|2008|pp=342-347|loc=§18.1}}
:<math>\limsup_{n\to\infty}\frac{\log d(n)}{\log n/\log\log n}=\log2.</math>
On the other hand, since [[Euclid's theorem|there are infinitely many prime numbers]],{{sfnp|Hardy|Wright|2008|pp=342-347|loc=§18.1}}
:<math>\liminf_{n\to\infty} d(n)=2.</math>

In [[Big-O notation]], [[Peter Gustav Lejeune Dirichlet]] showed that the [[Average order of an arithmetic function|average order]] of the divisor function satisfies the following inequality:{{sfnp|Apostol|1976|loc=Theorem 3.3}}{{sfnp|Hardy|Wright|2008|pp=347-350|loc=§18.2}}
:<math>\mbox{for all } x\geq1, \sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}),</math>
where <math>\gamma</math> is [[Euler–Mascheroni constant|Euler's gamma constant]]. Improving the bound <math>O(\sqrt{x})</math> in this formula is known as [[Divisor summatory function#Dirichlet's divisor problem|Dirichlet's divisor problem]].

{{anchor|Robin's theorem|Robin's inequality|Grönwall's theorem}}

The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by: {{sfnp|Hardy|Wright|2008|pp=469-471|loc=§22.9}}
:<math>
\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\,\log \log n}=e^\gamma,
</math>

where lim sup is the [[limit superior]].  This result is '''[[Thomas Hakon Grönwall|Grönwall]]'s theorem''', published in 1913 {{harv|Grönwall|1913}}. His proof uses [[Mertens' theorems|Mertens' third theorem]], which says that:

:<math>\lim_{n\to\infty}\frac{1}{\log n}\prod_{p\le n}\frac{p}{p-1}=e^\gamma,</math>

where ''p'' denotes a prime.

In 1915, Ramanujan proved that under the assumption of the [[Riemann hypothesis]], Robin's inequality
:<math>\ \sigma(n) < e^\gamma n \log \log n </math> (where γ is the [[Euler–Mascheroni constant]])
holds for all sufficiently large ''n'' {{harv|Ramanujan|1997}}.  The largest known value that violates the inequality is ''n''=[[5040 (number)|5040]]. In 1984, Guy Robin proved that the inequality is true for all ''n'' > 5040 [[if and only if]] the Riemann hypothesis is true {{harv|Robin|1984}}.  This is '''Robin's theorem''' and the inequality became known after him.  Robin furthermore showed that if the Riemann hypothesis is false then there are an infinite number of values of ''n'' that violate the inequality, and it is known that the smallest such ''n'' > 5040 must be [[superabundant number|superabundant]] {{harv|Akbary|Friggstad|2009}}. It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for ''n'' divisible by the fifth power of a prime {{Harv|Choie|Lichiardopol|Moree|Solé|2007}}.

Robin also proved, unconditionally, that the inequality:
:<math>\ \sigma(n) < e^\gamma n \log \log n + \frac{0.6483\ n}{\log \log n}</math>
holds for all ''n'' ≥ 3.

A related bound was given by [[Jeffrey Lagarias]] in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:
:<math> \sigma(n) < H_n + e^{H_n}\log(H_n)</math>
for every [[natural number]] ''n'' &gt; 1, where <math>H_n</math> is the ''n''th [[harmonic number]], {{harv|Lagarias|2002}}.