Development of the Valiron–Levin theorem on the least possible type of entire functions with a given upper $\rho$-density of roots

An entire function such that its roots have a given $\rho$-density and are located in an angle or on a ray is considered. For such a function, we solve the problem on the least possible type at order $\rho$. The case without assumptions about the location of the roots was considered by Valiron; the corresponding problem was completely solved by Levin.