I'm trying to find an explicit upper bound bound (explicit => without big O or small o notation, and with explicitly calculated constants) for the class number of a quadratic field.
So far, I've found that It was Littlewood who first addressed the question of how large the class number $h$ of an imaginary quadratic field $\mathbb{Q}(\sqrt{d})$ can be as a function of $|d|$ as $d \rightarrow$ $-\infty$ through fundamental discriminants. In 1927 he showed, assuming the generalized Riemann hypothesis (GRH), that for all fundamental $d<0$ $$ h \leq 2(c+\mathrm{o}(1))|d|^{\frac{1}{2}} \log \log |d|, $$ where $c=e^\gamma / \pi$, where $\gamma$ is Euler's constant.
Is there any way to get rid of the $o(1)$ in this expression? Or is there a more explicit bound available somewhere?
