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Let f, g be two distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 for the full modular group Γ = S L (2 , Z) , respectively. In this paper, we are interested in the upper bounds for the integral mean square of error terms involving λ f 2 i 1 (n) , λ f 2 (n j 1) and λ f 2 (n i 2) λ g 2 (n j 2) on average in terms of f, g respectively. Here i 1 ≥ 2 , j 1 ≥ 3 and i 2 , j 2 ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

Suppose that H ∗ is the set of all primitive cusp forms f of even integral weight k ≥ 2 for the full modular group S L 2 (Z) . In this paper, we establish asymptotic formulas for the second moment of Fourier coefficients of the triple product L-function L (s , f ⊗ f ⊗ f) and the related L-function L (s , sym 2 f ⊗ f) attached to f on average, which improves previous results. [ABSTRACT FROM AUTHOR]

Let f, g be two distinct normalized primitive holomorphic cusp forms of weights k 1 , k 2 and levels N 1 , N 2 , respectively. Denote by λ f (n) and λ g (n) the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the first change of sequences { λ f (n i) } n ∈ N and { λ f (n j) λ g (n l) } n ∈ N with positive integers i ≥ 2 and j , l ≥ 1 . And we also derive the lower bounds for the sequences { λ f (n i) } n ∈ N , { λ f (n j) λ f (n l) } n ∈ N of the same signs in the interval [1, x], which remains the best possible results in terms of order of magnitude, where i ≥ 2 , j , l ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

In this article, we investigate the sign changes of the sequence of coefficients at sum of two squares where the coefficients are the Fourier coefficients of the normalized Hecke eigen cusp form for the full modular group. We provide the quantitative result for the number of sign changes of the sequence in a small interval. [ABSTRACT FROM AUTHOR]

We prove nontrivial estimates for linear sums of squares of Fourier coefficients of holomorphic and Maass cusp forms twisted by additive characters. For holomorphic forms f, we show that if | α - a / q | ≤ 1 / q 2 with (a , q) = 1 , then for any ε > 0 , ∑ n ⩽ X λ f (n) 2 e (n α) ≪ f , ε X 4 5 + ε for X 1 5 ≪ q ≪ X 4 5. Moreover, for any ε > 0 , there exists a set S ⊂ (0 , 1) with μ (S) = 1 such that for every α ∈ S , there exists X 0 = X 0 (α) such that the above inequality holds true for any α ∈ S and X ⩾ X 0 (α). A weaker bound for Maass cusp forms is also established. [ABSTRACT FROM AUTHOR]