A Note on the Linnik's constant

Abstract

Let $P(a,q)$ be the least prime in the arithmetic progression $\{n\equiv a(modq)\}$. In this note, when $q$ has bounded cubic part and $(a,q)=1$, we combine the Heath-Brown's method and the Burgess's bounds for L-functions to obtain $P(a,q)\ll q^{4.5}.$