A large sieve inequality of Elliott-Montgomery-Vaughan type for Maass forms with applications to Linnik's problem (Q2637179)

This paper deals with a large sieve inequality of Elliott-Montgomery-Vaughan type over short intervals for Hecke eigenvalues of primitive Maass forms. An upper bound for the quantity \[ \sum\limits_{T-G\leqslant t_j\leqslant T+G}\Big{|}\sum\limits_{P<p<Q}\frac{b_p\lambda_j(p^\nu)}{p}\Big{|}^{2k} \] is the main result of the paper. Here \(1\leqslant G\leqslant T\), \(2\leqslant P<Q\leqslant 2P\), \(\nu\geqslant 1\) is a fixed integer, \(k\geqslant 1\) be an even integer, \(\lambda_j(n)\) be Hecke eigenvalues of primitive Maass forms, \(0<t_1\leqslant t_2\leqslant \ldots \) be real parameters related with construction of \(\lambda_j(n)\) and \(\{b_p\}\) be a sequence of real numbers indexed by prime numbers such that \(|b_p|\leqslant B\) for some constant \(B\) and all primes \(p\). Using the obtained estimate, the author presents two results on the generalized Linnik's problem.