How to prove $D(n)<2n(\log\log n)$?

Question

How to prove $D(n)<2n(\log\log{n})$ for all sufficiently large $n$ where $D(n)$ is the Divisor summatory function.

Answer 1

There is an unconditional result of Robin (1984) that, for integers $n \geq 13,$ $$ \sigma(n) < e^\gamma \; n \, \log \log n + \frac{ 0.6482316\ldots \; n}{\log \log n} $$ with the constant $0.64...$ chosen for equality at $n=12.$

From Robin's result, the version with coefficient $2$ is true for $n \geq 268.$

Note the earlier result of Gronwall (1913) in Hardy and Wright, Theorem 323 on page 266 of the fifth edition $$ \limsup \frac{\sigma(n)}{n \log \log n} = e^\gamma. $$ The proof is delayed until page 353, section 22.9, just after Merten's Theorem in section 22.8.

So, your statement is true. Not sure how you are going to prove it, you do not indicate your background or the source of the question.