The sum of divisors function and the Riemann hypothesis

Let $$\sigma (n)=\sum _{d\mid n}d$$ be the sum of divisors function and $$\gamma =0.577\ldots $$ the Euler constant. In 1984, Robin proved that, under the Riemann hypothesis, $$\sigma (n)/n < e^\gamma \log \log n$$ holds for $$n > 5040$$ and that this inequality is equivalent to the Riemann hypothesis. Under the Riemann hypothesis, Ramanujan gave the asymptotic upper bound $$\begin{aligned} \frac{\sigma (n)}{n}\leqslant e^\gamma \Big (\log \log n- \frac{2(\sqrt{2}-1)}{\sqrt{\log n}}+S_1(\log n)+ \frac{\mathcal {O}(1)}{\sqrt{\log n}\log \log n} \Big ) \end{aligned}$$ with $$S_1(x)=\sum _\rho x^{\rho -1}/(\rho (1-\rho ))= \sum _\rho x^{\rho -1}/|\rho |^2$$ where $$\rho $$ runs over the non-trivial zeros of the Riemann $$\zeta $$ function. In this paper, an effective form of the asymptotic upper bound of Ramanujan is given, which provides a slightly better upper bound for $$\sigma (n)/n$$ than Robin’s inequality.