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This paper studies explicit and theoretical bounds for several interestingquantities in number theory, conditionally on the Generalized RiemannHypothesis. Specifically, we improve the existing explicit bounds for the leastquadratic non-residue and the least prime in an arithmetic progression. We alsorefine 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 imaginaryquadratic fields. Finally, we improve the best known theoretical bounds for theleast quadratic non-residue, and more generally, the least $k$-th powernon-residue.

Conditional bounds for the least quadratic non-residue and related problems