On the least prime in an arithmetic progression (d'apres T. Xylouris)

Let a and q be two natural numbers with (a, q) = (1). By a classical theorem of J.P.G.L.Dirichlet,
there is a rational prime p and an integer l such that p = ql + a. No upper estimate for l in terms
of q was known, however, until the celebrated work of Yu.V.Linnik, who proved in 1944 that
$l < C q^L$ for some effectively computable constants C and L. In 1957  Ch.D.Pan proved
that one can take L=10000. The admissible value of L was improved in the works of several
authors. In 1992 D.R.Heath-Brown proved that the value L = 5.5 is admissable; in
his recent Ph.D. Dissertation T.Xylouris improves this to  L = 5. I intend to survey a few
ideas and methods in this area of the analytic theory of numbers.