A large sieve inequality of Elliott–Montgomery–Vaughan type for Maass forms with applications to Linnikʼs problem

In this paper, we establish a large sieve inequality of Elliott–Montgomery–Vaughan type over short intervals and explore its applications to Linnikʼs problem for Hecke eigenvalues of primitive Maass forms. One major obstacle is that the generalized Ramanujan conjecture is still unknown for primitive Maass forms and a generalization of Sarnakʼs result on the number of exceptional Hecke eigenvalues will get around this difficulty.