  This paper studies explicit and theoretical bounds for several interesting
quantities in number theory, conditionally on the Generalized Riemann
Hypothesis. Specifically, we improve the existing explicit bounds for the least
quadratic non-residue and the least prime in an arithmetic progression. We also
refine the classical conditional bounds of Littlewood for $L$-functions at
$s=1$. In particular, we derive explicit upper and lower bounds for $L(1,\chi)$
and $\zeta(1+it)$, and deduce explicit bounds for the class number of imaginary
quadratic fields. Finally, we improve the best known theoretical bounds for the
least quadratic non-residue, and more generally, the least $k$-th power
non-residue.
