A note on the least prime in an arithmetic progression

Let k, l denote positive integers with (k, l) = 1. Denote by p(k, l) the least prime p ≡ l(mod k). Let P(k) be the maximum value of p(k, l) for all l. We show $\text{lim inf}\text{P(k)}\text{(?(k)}\text{log}\text{k)}\text{≥ e}{}^{\mathrm{\gamma }}\text{= 1.78107…}$, where γ is Euler's constant and ? is Euler's function. We also show $\text{P(k)}\text{(?(k)}\text{log}\text{k)}\text{→ ∞}$ for almost all k.