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Let f be a normalized Hecke eigenform of weight k for the full modular group S L 2 (ℤ). In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg L-functions R (s , f × f) , supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant D < 0. We improve previous results in the case when the reduced form is given by (x 1 , x 2) = x 1 2 + x 2 2 . [ABSTRACT FROM AUTHOR]

Let f ∈ S k (Γ 0 (N)) be a normalized Hecke eigenforms of integral weight k and level N ≥ 1 . In the article, we establish the asymptotics of power moment associated to the sequences { λ f ⊗ f ⊗ f (Q (x ̲)) } Q ∈ S D , x ̲ ∈ Z 2 and { λ f ⊗ sym 2 f (Q (x ̲)) } Q ∈ S D , x ̲ ∈ Z 2 where S D denotes the set of inequivalent primitive integral positive-definite binary quadratic forms (reduced forms) of fixed discriminant D < 0. As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences. [ABSTRACT FROM AUTHOR]

In this article, we establish an average behaviour of the Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant D < 0 when the class number h (D) = 1 . We also obtain a quantitative result for the number of sign changes of the sequence of Fourier coefficients a(n) of the Hecke eigenforms, where n is represented by any primitive integral positive definite binary quadratic form of fixed discriminant D < 0 when the class number h (D) = 1 in the interval (x, 2x], for sufficiently large x. [ABSTRACT FROM AUTHOR]

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights k 1 and k 2 for the full modular group Γ = S L (2 , Z) , respectively. Denote by λ f (n) and λ g (n) the nth normalized Fourier coefficients of f and g, respectively. And set Q (x) a primitive integral positive-definite binary quadratic form of fixed discriminant D < 0 with the class number h (D) = 1 . In this paper, we establish a lower bound for the analytic density of the set { p : p = Q (x) for some x ∈ Z 2 , λ f (p i) λ f (p j) < λ g (p i) λ g (p j) } , where j ⩾ 1 , 0 ⩽ i ⩽ j are any fixed integers. Furthermore, we also consider the similar problem concerning the linear combinations of λ f (p j) and λ g (p j) in a given interval supported on the binary quadratic form Q (x) . Similar problems for triple product L-functions are also studied. [ABSTRACT FROM AUTHOR]