Let E be a CM elliptic curve defined over the rational numbers. In light of the Lang-Trotter conjecture, there is a question asking for an asymptotic formula for the number of primes p<x for which the reduction modulo p of E is cyclic. This question can be seen as an elliptic analogue of the prime number theorem and has been studied by Akbary, Cojocaru, Gupta, M.R. Murty, V.K. Murty, and Serre. In this talk, we will discuss their work and a short interval variant of such "cyclicity problem" which, in particular, presents a generalisation of the work of Huxley on the prime number theorem in short intervals.