I am given the harmonic series, $H_n$. How can I show that for $n\geq 1$, $$\sigma(n)\leq H_n+e^{H_n}\ln{H_n}$$
By the way, if you're not familiar with it, $\sigma(n)$ is the sum of all positive divisors of $n$.
I have no idea where to start.
2026-04-07 01:51:41.1775526701
Solve for bound of $\sigma(n)$ from harmonic series.
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In case this was a question in good faith:
since this problem is equivalent to the Riemann Hypothesis, don't expect to actually prove it.
LAGARIAS
For the curious, the best of the elementary RH versions to experiment with is the first one, by Jean-Louis Nicolas (1983). It says that RH is equivalent to the conjecture that, for all primorial numbers $P,$ $$ \frac{P}{\varphi(P)} > e^\gamma \log \log P $$ The point is that it is not difficult to get a computer to compute both sides of this for very large $P.$ The criterion of Lagarias and that of Robin (student of Nicolas) depend on working with colossally abundant numbers, which take real work.