A result by Lagarias is that the Riemann Hypothesis is equivalent to:
$$ \sigma(n) \leq H_n + \exp(H_n)\log(H_n)$$
Where $\sigma(n)$ is the sum of the divisors of $n$, and $H_n = \sum_{k=1}^n \frac1k$. I was curious if the above inequality is known to hold if we replace $H_n$ with $\log(n)$, or possibly $\log(n) -\gamma$, where $\gamma$ is the Euler-Mascheroni constant.