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

A large sieve inequality of Elliott–Montgomery–Vaughan type for Maass forms on GL($n,\mathbb R$) with applications

“…In this paper, we generalize the large sieve inequalities of EMV type to Maass forms on GL(n, R) for all n ≥ 3, and the result is comparable to the case of automorphic forms on GL(2, R) (see [8,15]). Our main tool is the automorphic Plancherel density theorem -a recent great progress due to Matz and Templier [10].…”

“…This device, known as the large sieve inequalities of Elliott-Montgomery-Vaughan (EMV) type, was generalized to the setting of primitive holomorphic cusp forms on GL(2, R) and applied to obtain some statistical results on Hecke eigenvalues of primitive holomorphic cusp forms in [8]. Later, Wang [15] generalized the results to the case of Maass forms on GL(2, R).…”

“…for t ≥ t 0 and log ε ≤ r ≤ (9 − ε) log 2 t. Remark 1.3. Refer to [8] and [15] for the case of GL(2, R).…”

A large sieve inequality of Elliott–Montgomery–Vaughan type for automorphic forms on GL$_3$

“…This large sieve inequality has been applied to get many statistical results on Fourier coefficients of primitive cusp forms (see [7], [10], [13]). In 2014, Wang [20] generalized this kind of large sieve inequality to primitive Maass forms on GL 2 . Motivated by the above works, we generalize Lau-Wu's work to Maass forms on GL 3 .…”