On comparing Hecke eigenvalues of cusp forms

Some Density Results on Sets of Primes for Hecke Eigenvalues

“…By comparison, ((1 + i − δ 1 )/4(ij + j + δ 2 ) 2 ) � (1/16[(j + 1)/2] 2 ) with i � 0, 1 ≤ j ≤ 8, and ((1 + i − δ 1 )/4(ij + j + δ 2 ) 2 ) � (1/4j (j + 1) 2 ) with 1 ≤ i � j ≤ 4. Hence, eorem 2 also generalizes previous results of Lao (see eorems 1 and 2 in [17]).…”

“…e next two lemmas are generalizations of Lemmas 1 and 2 and Lemmas 3.1 and 3.2 of Lao [17], respectively.…”

“…In this section, we will briefly recall some fundamental facts about primitive automorphic L-functions and give the main tools and definitions. For more details to learn automorphic L-functions, refer [17,[20][21][22][23][24][25][26][27][28].…”

“…Most recently, Lao [17] further studied Chiriac's questions [15] by considering the Hecke eigenvalues at prime powers. She established that the sets p|λ f (p j ) < λ g (p j ) with 1 ≤ j ≤ 8 and p|λ 2 f (p j ) < λ 2 g (p j ) with 1 ≤ j ≤ 4 have analytic density at least (1/16[(j + 1)/2] 2 ) and (1/4j(j + 1) 2 ), respectively.…”

“…2ab(a + b) p c 1 λ f p j + c 2 λ g p j p s + (a + b) 2 + 2ab p c 1 λ f p j + c 2 λ g p j 2 p s − 2(a + b) p c 1 λ f p j + c 2 λ g p j 3 p sFor the first sum in (67), by using (15), we getp c 1 λ f p j + c 2 λ g p j p s � O(1), as s ⟶ 1 + . (68)For the second sum in (67), one can easily obtain from (15),(17), and (50) that p c 1 λ f p j + c 2 λ g p j 2 O(1), as s ⟶ 1 + . (69)…”