A conditional bound for the least prime in an arithmetic progression

Assuming the generalized Lindelöf hypothesis for Dirichlet $L$-functions, we establish that the least prime $p\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}q)$ satisfies $p{\ll }_{\epsilon }{q}^{2+\epsilon }$. This achieves a bound that nearly matches the classical estimate implied by the generalized Riemann hypothesis.